Mourinho leapt from his seat on United's bench to confront Marco Ianni after Sarri's assistant coach celebrated Ross Barkley's dramatic late equaliser for Chelsea by pumping his fists in front of the away dugout at Stamford Bridge.
Sarri admitted Ianni was in the wrong and apologised to his opposite number.
The former Inter boss confirmed he had accepted an apology from the assistant after the 2-2 draw, while Sarri said he would deal with the matter.
Mourinho, who guided Chelsea to three Premier League titles, was prepared to move on from the incident, although he was keen to absolve himself of any blame. If somebody tells me before the game "You go to Stamford Bridge you'll get a point", you say "Okay, that's good". I think it's finished but I want to be sure. "I have spoken with the member of my staff, and I again spoke with Mourinho to say sorry to him". I think he's developing.
"We are talking about a coach that has won everything, everywhere, so I think you have to respect him". "But I also made mistakes in football matches and I will make more".
"When we scored the second goal there was a little bit of a discussion outside the pitch".
He said on the BBC: "Whatever Mourinho said to his team (at half time) totally changed them".
Mourinho said after that match that: "I am not annoyed with anything".
Antonio Rudiger gave the Blues a deserved lead in the first-half, but United came roaring back into the game after the break. "I want to thank Chelsea for its honesty too - but I am not happy that it is going too far with the young boy", the Man United manager was quoted as saying by Sky Sports.
"I would really like the boy not to go through more than that, let him work".
"Then after 1-1, we didn't play our football, only the long ball, we are not organised for the second ball".
"End of story with me, is the minute he apologised to me".