Southampton attempted to respond immediately.
Koeman shouted towards his midfielders, demanding that they move the ball faster and avoid giving Liverpool another opportunity to press them into mistakes.
Tadić received on the right and passed inside to Davis, who looked towards Pellè.
The striker attempted to hold the ball with his back to goal.
Sakho challenged him.
Pellè managed to lay the ball off towards Mané, who was arriving from the left.
The winger took a touch and accelerated towards the penalty area.
Clyne retreated, attempting to show him towards the touchline.
Mané shifted the ball onto his right foot.
Can moved across to cover.
Mané attempted a shot from the edge of the box.
The ball travelled towards the far corner, forcing Mignolet to dive.
The goalkeeper reached it with both hands and pushed it away.
"Good save!" Skrtel shouted.
The rebound rolled towards the edge of the area.
Azim had already recovered into the central channel.
He approached Mané from the side, preventing the winger from reaching the loose ball cleanly.
Mané attempted to recover possession.
Azim stepped across and poked the ball towards Lucas.
The midfielder collected it and cleared towards Can.
Liverpool had survived the attack.
Mané turned towards Azim.
"You're a striker. Stop following me."
Azim looked at him.
"Stop giving me reasons."
Mané shook his head and moved away.
The exchange was brief, but Azim felt a quiet satisfaction. He had helped score another goal, yet he had also returned to the defensive work immediately afterward.
The 4-0 scoreline had not changed his responsibilities.
If Southampton were allowed to attack freely, they could still create opportunities. Mané had already shown that he could threaten Liverpool's defensive line with pace and direct running.
Azim needed to help prevent those attacks from developing.
He was not simply waiting for the next chance to score.
He was helping Liverpool control the match.
The next 10 minutes became a contest over the spaces between Southampton's midfield and defence.
Koeman's players were more cautious with their passing, but they could not afford to abandon their attacking intentions. Their full-backs continued to advance whenever possible, while Davis moved around Lucas in search of a pocket of space.
Liverpool responded with fluid movement.
Firmino dropped towards midfield.
Coutinho moved inside.
Milner offered width on the right.
Azim drifted between the central channel and the left side, constantly testing Van Dijk's willingness to follow him.
At 58 minutes, Liverpool recovered possession through a coordinated press.
Fonte passed towards Clasie.
Azim approached from the front, blocking the route into Wanyama. Firmino moved across to close Fonte's return option, while Coutinho advanced towards Cédric.
Clasie attempted a pass into Davis.
Lucas stepped forward and intercepted it.
The ball rolled towards Can, who immediately passed to Milner.
Milner looked forward.
Azim had begun moving towards the right channel, drawing Van Dijk away from the centre.
Firmino ran beyond Wanyama.
Coutinho moved towards the left side of the penalty area.
Milner played into Firmino, who controlled the ball and turned towards goal.
Wanyama challenged.
Firmino passed backwards to Can.
The midfielder struck first time.
Fonte blocked the shot.
The ball deflected towards Coutinho, who attempted to recover it near the byline.
Cédric stepped across and forced him to retreat.
Liverpool retained possession, but the attack lost its immediate danger.
Can looked towards the penalty area.
"Sorry! I should have placed it!"
Milner shouted back.
"Next time, make sure you get the shot away!"
Azim had been moving towards the far post, ready to attack a cross.
When the ball went out of the immediate shooting position, he turned and began pressing again.
The pattern repeated itself.
Southampton tried to build.
Liverpool closed the passing lanes.
The ball moved towards the touchline.
A hurried pass followed.
Liverpool recovered possession.
The home side were not winning every duel, but they were controlling where most of those duels occurred.
And that was becoming increasingly difficult for Southampton to change.
At 62 minutes, the 3rd goal of the 2nd half arrived through another fluid combination.
Lucas intercepted a pass from Davis and played towards Can.
Can moved the ball into Coutinho, who had drifted inside from the left.
Coutinho took a touch towards the centre, drawing Wanyama away from the space near the penalty arc.
Firmino moved towards him.
Azim began his run behind the centre-backs.
Van Dijk followed.
Coutinho passed to Firmino, who immediately returned the ball into the central channel.
The exchange had forced Clasie to turn towards his own goal. Wanyama was recovering from the wrong side, and Fonte had moved towards Azim.
Firmino received the next pass with his body open towards goal.
He controlled the ball with his right foot, then shifted it onto his left as Stekelenburg advanced.
The goalkeeper expected a shot towards the far corner.
Firmino delayed for half a second.
Then he rolled the ball into the near side of the net.
Liverpool 5, Southampton 0.
Firmino spread his arms and ran towards the corner flag, laughing as Coutinho chased him.
Azim reached them first and jumped beside the Brazilian, driving his fist into the air towards the supporters.
"That's what happens when you run!" Azim shouted.
Firmino laughed.
"You are never going to let me forget that, are you?"
"Not while you're scoring."
The players gathered around Firmino.
Milner arrived, clapping his hands.
"Excellent! That's the movement. That's what we talked about!"
Klopp was beside the touchline, roaring with delight.
The commentators highlighted the combination that had produced the goal.
"That is excellent from Liverpool. Coutinho draws the midfielder away, Firmino combines with him, and suddenly Southampton's defensive line has to deal with 3 different runs."
"Look at Azim. He does not receive the final pass, but his movement forces Van Dijk towards the channel. That leaves Firmino with the room to finish. The forward has scored 3 goals, yet he is also making life easier for everyone around him."
Koeman called his players towards the touchline as they prepared to restart.
His expression was stern.
"Stay organised! You cannot continue giving them these spaces!"
Wanyama looked towards the manager.
"They keep moving. We cannot follow everyone."
"Then decide who follows and who covers!" Koeman replied. "Do not all chase the same player!"
The instruction was sensible.
But Southampton had already conceded 5 goals, and their players were struggling to balance defensive discipline with the need to create a response.
Van Dijk looked towards Azim.
The forward was laughing with Firmino.
For the 1st time that afternoon, the centre-back seemed less interested in provoking him than in understanding how to stop him.
Liverpool's next opportunity came from a mistake.
Southampton attempted to build through Bertrand, who received near the touchline and passed inside to Davis.
Azim began pressing from the central channel.
His approach blocked the route towards Wanyama, encouraging Davis to play backwards.
The attacking midfielder hesitated.
That hesitation allowed Can to step forward.
Davis attempted the pass anyway.
It was underhit.
Can intercepted it and immediately played towards Milner.
Milner carried the ball forward before finding Azim, who had dropped into the space between Southampton's midfield and defence.
Van Dijk moved towards him.
Azim controlled the ball with his right foot and turned away from the defender, using his body to protect possession.
Wanyama approached from the side.
Azim passed backwards to Firmino.
The Brazilian played first time towards Coutinho, who was moving into the left channel.
Coutinho took a touch and attempted to cut inside.
Cédric blocked the route.
The ball deflected towards Milner, whose first touch was slightly too heavy.
Fonte stepped forward and cleared.
The chance was gone.
Milner looked towards the ground.
"Too heavy."
Azim approached him.
"At least we got it back."
Milner glanced at him.
"That doesn't mean we can start giving it away."
Azim nodded.
The captain was right.
Liverpool's pressing had created another opportunity, but the attack had lost its momentum because of a poor first touch. If Southampton had recovered the ball cleanly, their attackers would have had space to counter.
The team could not rely on the opposition making mistakes forever.
They needed to keep their own decisions disciplined.
At 70 minutes, Southampton made a substitution.
Oriol Romeu replaced Clasie.
The change added fresh legs to central midfield and gave Koeman a chance to restore some control in front of the defence.
Romeu took his position beside Wanyama, while Davis remained ahead of them.
Southampton's shape became more compact.
Van Dijk stayed close to Fonte.
The full-backs were more cautious about advancing together.
The adjustment helped them regain some control of the central space, but it also left Pellè more isolated whenever they attacked.
Liverpool recognised the change.
Azim dropped deeper whenever the ball moved towards Lucas, forcing one of the Southampton midfielders to decide whether to follow him.
When Romeu stepped forward, Firmino moved into the space behind him.
When Wanyama stayed close to the centre-backs, Coutinho carried the ball inside.
Liverpool did not need to force the same pass repeatedly.
They could move the ball from one side to the other until Southampton's midfield was drawn out of position.
At 72 minutes, the visitors tried to build through Romeu.
The midfielder received from Fonte and turned towards Davis.
Azim began pressing from the side.
Romeu passed towards Wanyama, but Firmino had already moved into the passing lane.
The ball struck the Brazilian's foot and deflected towards Can.
Can recovered it.
He passed immediately towards Milner, who played forward to Azim.
The forward received near the halfway line and saw that Southampton's defensive line was retreating.
Firmino was running into the left channel.
Coutinho was moving towards the centre.
Can was advancing behind the ball.
Azim could have carried it forward himself.
Instead, he waited for Van Dijk to step towards him.
The defender moved forward to close the space.
Azim then played a short pass around the centre-back's outside shoulder, finding Can in the gap between the defensive line and midfield.
Can took a touch.
Fonte stepped across.
The German midfielder shifted the ball onto his right foot and struck from just outside the penalty area.
The shot travelled low through the crowd of defenders.
Stekelenburg moved across, but the ball passed beyond his reach and entered the bottom corner.
Liverpool 6, Southampton 0.
Can turned away, throwing both arms into the air.
Azim ran towards him and pulled him into an embrace.
"That's another one!" the forward shouted.
Can laughed.
"You finally passed to me!"
"I pass to everybody."
"Then keep doing it!"
Milner reached them, clapping with obvious satisfaction.
"Well done! Excellent decision!"
The commentators praised the attack.
"That is another goal created by Azim's movement and decision-making. He draws Van Dijk forward, releases the ball into the space behind and allows Can to attack the penalty area."
"And Can finishes it beautifully. Liverpool have scored 6, and Southampton are struggling to find any answer to the movement around Azim."
Klopp celebrated with another vigorous punch of the air, then turned towards his assistants.
The match had become a demonstration of the principle he had explained at half-time.
The ball moved.
The players moved.
The opponent had to choose.
And Liverpool were punishing the choices Southampton were forced to make.
Southampton responded with a more direct approach.
Pellè began positioning himself closer to Liverpool's centre-backs, while Mané looked for opportunities to run beyond Clyne.
Ward-Prowse, who had entered earlier, moved towards the right flank and prepared to deliver crosses.
Koeman knew the scoreline was beyond easy recovery.
But he did not want his team to finish the match without attempting to score.