Ianni celebrated Ross Barkley's stoppage-time equaliser for the Blues in front of the visiting bench at Stamford Bridge, much to the annoyance of Jose Mourinho.
The United manager had to be restrained by stewards as tempers flared in the aftermath to the late goal, though Sarri insisted he did not witness the incident.
Sarri has, however, spoken to the member of his staff who was involved, while Mourinho insisted there was no bad feeling after receiving an apology from both Sarri and Ianni.
"Sincerely I have not seen what happened, but then I spoke to Jose, of course, then I spoke to the member of the staff because I think that we were on the wrong side, so I have dealt with this immediately," Sarri told Sky Sports.
"I think it's private. I spoke with the member of staff and I have to be very hard. I cannot come here and tell everything, otherwise with my staff I am not credible.
"I am not really happy, as I said before, after I have spoken with Jose I understood that we were on the wrong side of the situation."
The Blues fought back to preserve Sarri's unbeaten Premier League record, substitute Barkley securing a point when he fired home in the sixth and final minute of additional time after David Luiz's flying header had hit the post.
Antonio Rudiger had headed the hosts in front, but United turned the game around, Anthony Martial grabbing a brace in an absorbing contest between Mourinho's current club and one of his former employers.
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.
"I want to play for three points. For 60 minutes we played our football," the former Napoli coach said in his post-match press conference.
"Then after 1-1, we didn't play our football, only the long ball, we are not organised for the second ball. In this type of football, United are better than us. I am really disappointed in this type of performance.
"The 2-2 in the last minute is okay, okay for the confidence of my players, but I have to look [at] something.
"I have seen the match on the bench, and now want to study the second half of the match."