The Dutchman's deflected injury-time free-kick earned United a 3-2 win at the Etihad Stadium after City had come back from 2-0 down.
Wayne Rooney netted twice before the break, with Yaya Toure and Pablo Zabaleta snatching goals for the hosts in the second half.
Van Persie said: "We had a really good start, because of Wayne's two really good, clinical finishes.
"We were in control but City always come back. But it's a really special way to end this game, with the three points in a really dramatic way."
Rooney said of his team-mate's winning goal: "We work on that on the training ground and I had no doubt he could score from there."