The loss saw Chelsea slip below Arsenal to sixth in the table on goals scored and raised further questions over the future of manager Maurizio Sarri.
Aguero, who scored a hat-trick seven days ago against Arsenal, has equalled Alan Shearer's record of 11 Premier League hat-tricks and is now joint top scorer in the league this season with Liverpool's Mohamed Salah on 17 goals.
When asked whether he fears being sacked, Sarri replied: "I don't know, you have to ask the club".
'I didn't see him at the time but I will go to say hello to him later, as usual, ' Sarri explained at his post-match press conference. With a cleverly taken free kick down the right, Kevin De Bruyne released Bernardo Silva, who cut back for Sterling to strike into the net.
"We had only to stay in the match but the reaction was not good. We're watching a really weak football team here".
Chelsea boss Sarri headed straight down the tunnel without shaking hands with Pep Guardiola, but the pair spoke privately after the game. Beyond that, it's 9/1 (10.0) that no Premier League boss leaves before the end of the season. People think if you buy players you can immediately come in and win, but you need time.
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"That's why I'm enjoying it in Manchester".
Insisting he was happy for Vardy to take the penalty, Puel said: "I gave him the opportunity. He doesn't believe I am good because I win and bad because I lose".
Jurgen Klopp's men maintain the advantage of a game in hand, but handed a reprieve by Liverpool's draws against Leicester and West Ham in recent weeks, City look determined to become the first side in a decade to retain the Premier League.
Guardiola fell to the floor in exasperation when Aguero somehow side-footed wide with his easiest chance of the game moments later. 'I have a great relationship with him, no problem'. "But his reaction was a little bit better". We started well but after four minutes we conceded a goal in a stupid way.
And Ilkay Gundogan's shot from the edge of the area then had too much power for the world's most expensive goalkeeper Kepa Arrizabalaga to compound Chelsea's woes.
The third for AGUERO would come in the second half through a penalty kick.
Sterling wrapped up a brilliant afternoon's work after substitute David Silva opened up the Chelsea defense yet again and Zinchenko picked out the winger to tap in from close range.