Earlier, Ross Barkley scored in the sixth minute of injury time at Stamford Bridge to level the match at 2-2 after Anthony Martial had scored twice in the second half to cancel out Antonio Rudiger's opener.
Yet, that is exactly what Ianni did to him after Barkley's goal, which sparked ugly scenes near the dug-out as Mourinho tried to "go after" the 36-year-old Italian.
The celebrations caused an altercation in front of the technical area before tensions were calmed before the final whistle.
Play was eventually able to resume after a further fracas in the centre circle of the pitch.
The FA said the Portuguese manager had been "formally reminded of his responsibilities" after the 2-2 draw.
But his frustration at blowing what could have been a huge win was obvious as the visitors remain seven points adrift of Chelsea, and could fall nine points behind Manchester City and Liverpool should they win their respective fixtures later Saturday.
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Television cameras captured Mourinho muttering what are purported to have been "abusive and/or insulting and/or improper" remarks.
The young coach apologized right away for his actions and so too did Sarri. I don't think he deserves more than what he got. I think I would try always to control myself and to respect the stadium and the supporters that were my supporters for many years.
Barkley, who has been in great form for Chelsea in recent weeks, went on to regret the decision of the arm tattoo sleeve - which included the date of his Everton debut - and later got them removed. "So if it's going in other directions, I also want to go in another direction".
"I'm happy to see my contract out to the very last day of that contract", Mourinho said Monday, "and I would like to stay after my contract ends".
Sarri was disappointed with how his team struggled to cope with the visitors' more direct approach in the second half, though he will watch the game back before making a more detailed assessment of Chelsea's performance.
Neville concluded: "If I was Maurizio Sarri I'd send that guy in and sack him from the club".