Two Van Persie penalties either side of a fortuitous Theo Walcott goal helped the Gunners erase first-half goals from Richard Dunne and Darren Bent and set up a last-16 meeting with Sunderland or Middlesbrough, but McLeish was unhappy with another incident to do with Arsenal's top scorer.
He felt the Dutchman had caught defender Carlos Cuellar as they jumped for a high ball in the 64th minute. He said: "I thought it was a clear elbow, but the referee did not seem to think there was anything in it."
McLeish added: "Would I like the FA to look at it? Yes, if it was an elbow in the face - look at my nose, it is a result of elbows all my career. It is something which is dangerous.
"I never mentioned it to the referee. I saw it in real time. I don't like to point the finger at players, but it has happened to mine enough times."
Should it transpire referee Mike Jones witnessed the incident at the time, then no retrospective action can be taken by the disciplinary authorities.
McLeish believes his side had given themselves every chance of progressing before Arsenal mounted their fightback.
He said: "I am very disappointed we did not see it out and it is a missed opportunity. The fans will be gutted at seeing us let that lead slip away. We have to learn from this and try to obtain that winning mentality."
Arsenal manager Arsene Wenger, meanwhile, praised the character of his side, who have lost three successive Barclays Premier League games to lose ground on the top four.
He said: "I was very impressed because I felt that when you come out of a bad period like we are, and you are 2-0 down, you can feel sorry for yourself or you can show you have the mental qualities to play for this club. I was very proud the team came out playing on all cylinders and showed they are not ready to lie down."