Although City started brightly, Wayne Rooney netted twice in the first half to set the visitors up with a two-goal advantage, but strikes from Yaya Toure and Pablo Zabaleta set up a grandstand finale between the rivals.
However, the Netherlands striker won the game with a 91st-minute free kick courtesy of a defelction off Samir Nasri - and the City boss admitted his side "are not in a good moment" as they fell six points behind United.
|9/2||Roberto Mancini is 9/2 with William Hill to be the next Premier League manager to leave his post|
"For 20 minutes they didn't touch the ball, they shot three times and they scored three goals. I think that a game shouldn't finish like this.
"When it was 2-0 we had chance after chance but we were unlucky. We played, they won, but also we showed we are better as a team.
"We are disappointed but it is because we lost a game we didn't deserve to lose. I can say that I am happy for our performance."
Mancini also bemoaned that his side slipped to a late loss, and believes his defence should have been more wary of a player as lethal in front of goal as Van Persie.
He continued: "I think in the last second we should pay more attention in this situation, because you're playing against a top player, and Van Persie shoots very, very well."
While City slipped six points behind league leaders United in the Premier League, the 48-year-old manager feels his side have enough time to retain their title.
He added: "We have time to recover, if we have the same attitude but without mistakes I think we have time to recover these six points.
"We are not in a good moment but I think we should restart from this performance and we should understand we are a good team.
"It was a good game until the last two seconds for our supporters."