City's demolition of the Blues was their heaviest defeat in 28 years and made it four away losses in a row in all competitions for Maurizio Sarri's side.
Sergio Aguero hit a hat-trick, Raheem Sterling scored twice and Ilkay Gundogan was also on target as City ran riot to reclaim top spot from Liverpool.
The Blues have now lost three straight Premier League away games by an aggregate 12-0 scoreline and defender Azpilicueta acknowledged they were far from good enough.
"It is one of the worst nights in my career. It is difficult to explain what happened," he told Sky Sports .
"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 [ a 2-0 victory in December ] 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."
The win puts Chelsea boss Sarri under mounting pressure at Stamford Bridge after he admitted last week that he struggles to motivate some of his players in west London.
Chelsea face Malmo next time out on Thursday before a clash with Manchester United in the FA Cup, and play City again in the Carabao Cup final at Wembley Stadium on January 24.
The Italian will be hoping he can turn the Blues' form around in those outings, especially with their next scheduled Premier League outing coming against third-placed Tottenham at the end of the month.