"I did not get respect back from Chelsea fans but that is not my responsibility", Mourinho said.
"And after that his assistant came to me in Sarri's office, and apologised to me".
United were on the verge of securing their first win at Stamford Bridge since 2012 after mounting an impressive comeback in the second half, with Anthony Martial scoring twice after Antonio Rüdiger had given Chelsea an early lead.
"Sincerely, I have not seen [what happened], but I have spoken with Jose [Mourinho], then I have spoken with a member of staff because I think we were on the wrong side, so I have dealt with it immediately", he said.
The coach jumped from his seat and was held back by those around, as he attempted to sort things out with Ianni.
"In fact I would say sometimes it's a reckless approach away from home, even at home".
Security from both sides pushed their way in and at one point Mourinho was temporarily immobilised by the bear-hug of one of Chelsea's biggest matchday stewards. What I did here today I will do in Madrid, in Milan, in Porto, the reaction from the fans is not up to me. We were in the wrong on the situation.
Chelsea snatched out a draw against Manchester United in a four-goal thriller on Saturday.
Yes, it wasn't his fault that Ianni celebrated so crazily in front of him, but there was little sense of Mourinho biting his tongue and refusing to take the bait.
Sarri said his side lost out to United in the physical battle as the game wore on.
"He is a great manager, we have that freedom on the pitch, and when we don't have the ball, we know what to do so it is quite simple". Sometimes the opponent is better than us. I can not come here and tell everything, otherwise with my staff I am not credible. I don't want the long ball, I don't want the second ball. "I want to play with short passes - stop!"