"My feeling is that we work on the pitch", Mourinho said.
It appeared that Chelsea were facing their first defeat of Sarri's reign, before Ross Barkley popped up inside the area to score an equaliser.
Chelsea fell behind thanks to an Anthony Martial brace but Barkley fired home a stoppage time equaliser.
Mourinho reacted furiously to the gesture and raced after him up the touchline.
He told Sky Sports: 'I am not annoyed with anything. But I also made mistakes in football matches and I will make more so when after the game he [Ianni] came to me to apologise, I accepted his apologies and nothing more to say. It's private. I can not come here and tell anything otherwise with my staff I am not credible. but I am not really happy about it. "'Forget it, because I've also made lots of mistakes in my career".
Chelsea coach Maurizio Sarri backed Mourinho, saying he had already reprimanded his staff member for his role in the incident.
It was a dramatic conclusion as Mourinho saw victory snatched away after he had replied to the taunts of Chelsea fans by holding up three fingers to remind them of the three Premier League titles he won while in charge at Stamford Bridge.
Questioned after the game, the Portuguese said it was in the past.
"I did not get respect back from Chelsea (fans) but that is not my responsibility".
That may be interpreted as implied criticism of Mourinho, but Sarri was not about to emulate his predecessor, Antonio Conte, in igniting a public feud with the former Porto, Chelsea, Inter Milan and Real Madrid boss.
The Red Devils will return to Champions League action on Tuesday night when they host Juventus at Old Trafford in the group stage.
Manchester United manager Jose Mourinho has been no stranger to criticising his own team. I understood that we were in the wrong side of the situation. So, I have to be myself, always, otherwise I am not credible.
Manchester United coach Jose Mourinho has promised to be on his best behavior when he takes his side to play Chelsea in Stamford Bridge on Saturday lunchtime.
The United manager jumped from his seat and hurled himself towards the member of staff, clearly enraged, and a huge melee ensued by the tunnel, with staff, players and security all getting involved.
Chelsea's players have flourished under Sarri, none more so than the Premier League's top goalscorer Eden Hazard.