The heavy defeat at the Etihad Stadium was the club’s worst result since the Premier League began in 1992.
“It is one of the worst nights in my career,” Azpilicueta, who gave away the penalty for Sergio Aguero’s third goal, said.
“It is difficult to explain what happened.
“The first five or six minutes went well but that is not enough. It is very frustrating and disappointing. We are conceding a lot of goals and we cannot accept that.
“When you lose 6-0 we have to accept we didn’t play well and made errors. We knew it would be tough.
“When we won at Stamford Bridge it was because we played well and were together. Today was not the case.
“All I can do is apologise to the fans because it is not acceptable.”
Aguero hit a hat-trick, Raheem Sterling grabbed a brace while Ilkay Gundogan was also on target in the rout.
The three points see Manchester City leapfrog Liverpool at the top of the Premier League table but Jurgen Klopp’s men have a game in hand.
Sterling, who was at his electric best, said: “It was a terrific performance from the team – the manager set us up and got us ready for the game.
“Sergio is a player we depend on. You saw with his first strike putting the ball in the top corner – you have to cherish players like that.
“It was just a win and another three points. The most important thing is to get three points.”
Aguero missed an open goal from a yard out before netting three times.
Discussing the blunder, he said: “I don’t know, sometimes it happens. The next one I scored. The important thing is to get the ball and for the team to win.”
Aguero’s treble also saw the Argentinian equal Alan Shearer’s record for Premier League hat-tricks.
“I have to continue the same way, but it’s important to win,” he said.