The Blues defender was the last of many players including Eden Hazard, David Luiz, Cesar Azpilicueta and Marcos Alonso to go to apologise to the travelling support on Sunday.
Stewards at the Etihad Stadium stepped in to pull away from the supporter, who was shouting at the Chelsea centre back, as he handed his shirt over.
Rudiger confronted the angry fan by saying: “I’ve come over to say sorry - stand in front of me if you‘ve something to say.”
The scene, which has been both filmed and pictured, came after Chelsea suffered their heaviest defeat for 28 years and as they drop to sixth place, the lowest position that they have yet registered this season.
The majority of the away section chanted "Rudi, Rudi, Rudi" after seeing his passion in what was an emotional moment for those at the club.
Manchester City's huge win helped propel them back above their main title rivals Liverpool, as Sergio Aguero's hat-trick, a Raheem Sterling brace and Ilkay Gündogan's long-range effort humbled the Stamford Bridge club.
Chelsea's vice-captain Cesar Azpilicueta was one of the few to face the media after the painful result, and he used the opportunity to apologise to fans.
"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.
Rudiger staying to apologise to Chelsea fans.— Lewis (@CarefreeLewisG) February 10, 2019
At least he cares pic.twitter.com/lmLyEprK1c
"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."