United was headed for a stunning and uplifting win when, in the 96th minute, Ross Barkley slammed home a rebound to level the match at 2-2.
The Portuguese coach has lost on each of his three visits to former club Chelsea since talking over at Old Trafford in 2016.
Sarri, who is yet to win a trophy in his management career, said: "The results speak for him".
The manager tried to get up-close with the air-punching coach, yet he was swiftly restrained by stewards. Staff from both teams clashed in the tunnel and Mourinho later revealed he had accepted an apology about the flashpoint from his opposite number after the game.
Given the circumstances, it was nearly a surprise that Mourinho calmed down quickly enough to shake Sarri's hand at full-time.
Eddie Howe's side are in sixth place with five wins already this season and are chasing a third straight Premier League victory for the first time since March 2016.
Mourinho also confirmed via the BBC that Ianni had also apologised for his actions.
Mourinho goes back to Chelsea's home ground in charge of Manchester United and still under pressure following a poor start to the season.
"I think [it was] a fantastic game with a fantastic result for Chelsea and an bad result for us", he said. I think it is finished. The ball made its way to Marcus Rashford and he played the ball across to Martial, who took one touch and curled a shot into the bottom corner. "We were in the wrong". So I have dealt with this immediately.
Mourinho conceded that he was upset at the way in which Ianni celebrated in front of him.
The former Blues boss has not won at his former ground since taking over at United and he was involved after the home bench erupted in celebrations at the late leveller. I accept. For me, the story is over.
The Daily Mirror ran a back page story on the morning of United's last Premier League game against Newcastle claiming Mourinho would be sacked that weekend regardless of the result against Rafael Benitez's side.
"After the goal to make it 1-1 we didn't play our football, only the long ball".
"We controlled that very, very well". But I have spoken with Jose and the staff because I think we were on the wrong side. He said: "We are not organised like this". He was for the first time best player in the Premier League with me, so I think the feelings are good. With this kind of football, United are better than us.