Fulham lost the encounter 3-1 at Old Trafford and tensions boiled over when Willian was sent off for a deliberate handball on the goal line.
Silva was sent off for dissent and Mitrovic swiftly followed his manager for an early bath when he pushed referee Chris Kavanagh.
The Fulham boss has been charged by the FA for misconduct and Mitrovic could face an additional punishment aside his three-game ban.
"I have been away with my national team and have had some time to reflect on the Manchester United match," Mitrovic told the club's website.
"As everyone knows, it was a very emotional one, in which the team gave absolutely everything on the pitch to try to reach the semi-final of the FA Cup.
"On a personal level, I regret my actions that led to me being sent off. I allowed my frustration to get the better of me, and how I reacted was wrong.
"I have accepted the three-match ban for my red card, I have spoken to Chris Kavanagh to apologise, and I have volunteered to accept a club fine.
"I now want to do what I can to put this incident behind me and get back to helping my teammates on the pitch as soon as possible."
Silva echoed his player's sentiments, adding: "Emotions were high at Old Trafford and, as I said in my press conference after the game, I should have controlled my own emotions better.
"It was a very difficult moment, in a match that we had dominated, but I could have handled the situation better.
"I regret what happened, and I have spoken to Chris Kavanagh to apologise.
"He knows that I respect him and the job he has to do. I am sorry that I was not on the pitch to try to lead the team to Wembley, but I am thankful for all the support I have received from Fulham fans over the last few days."
