Ex-Chelsea boss Mourinho was enraged by Ianni after he celebrated wildly in front of him following Ross Barkley's late equaliser at Stamford Bridge.
Ianni jumped off his seat as the ball went in and celebrated in the face of a very bemused Mourinho.
His clenched fist provoked the United boss' ire, sparking a mini-riot at the Bridge which overshadowed a fine comeback from the hosts.
Mourinho took off after his rival and had to be held back as a mass brawl broke out at the entrance to the Chelsea tunnel.
Backroom teams from both sides, along with security officials, were caught up in the tussle before players also got involved.
Who is Marco Ianni?
Ianni was appointed by Chelsea in early August, joining a team that included former Chelsea players Hilario, Carlo Cudicini and Gianfranco Zola.
Ianni, who was born in 1982, retired from football at the age of 32 and immediately turned his attention to coaching.
He spent two years on the staff of Roberto Cappellacci in his native Italy - first with Cosenza and then Campobasso - before joining Sarri at Napoli, where Ianni worked as the club's technical assistant.
In a recent interview, Ianni revealed he had "a good professional and human relationship" with Sarri.
He will hope that ensures he avoids serious punishment after the Chelsea coach stressed he had to be "very hard" on Ianni for the incident with Mourinho.
How did Mourinho react to Ianni's actions?
Despite the fracas, Mourinho attempted to play down the incident once tempers had cooled post-match.
"Oh come on, I can tell you that 97 minutes of the game was so good that you have to focus on that," he said to Sky Sports.
"I did not get respect back from Chelsea [fans] but that is not my responsibility. What I did here today I will do in Madrid, in Milan, in Porto, the reaction from the fans is not up to me."
The United boss subsequently revealed that he had received an apology from Ianni.
"I'm not annoyed with anything," said the Portuguese. "What happened was with Sarri's assistant. Sarri was the first one to come to me and say he would resolve the problem internally.
"After that his assistant came to me in Sarri's office and apologised to me. I immediately told him 'if you really feel that way and you want to apologise, of course I accept apologies and forget it because I also made lots of mistakes in my career, so I'm not going to kill because of one'."
How did Chelsea react?
Like Mourinho, Chelsea were keen to put Ianni's antics behind them once the final whistle had blown.
"I don't know [what happened]," Antonio Rudiger told Sky Sports. "I just saw the benches standing up, and I was like 'what, a fight?'.
"But it is emotions, it's just football. I didn't see anything on the pitch. But that's football.
"At the end of the day you shake hands. The game was still going on [when it happened]. That's emotion, at least everything has calmed down now."