"It's very hard to lose a derby, but even more so in the last minute, so I'm very disappointed," Mancini said. "We dominated the second half, and for this reason I am really disappointed.
"They scored from their first two goal-scoring opportunities but after that we dominated the game and fought back really well -- but we can't then give a goal away after working so hard.
"I think we made a mistake for the Van Persie free-kick at the end by only putting three men in the wall. I wanted four but I couldn't get the instruction over and I think that cost us the goal. We should have defended this free-kick better and we had chances to win this game, but there is nothing we can do about it now."
United had conceded first in 15 of 23 games so far this season, but a double inside the opening half hour from Wayne Rooney -- which made the 27-year-old England striker the youngest player to score 150 in the EPL -- put Ferguson's team in the driving seat.
United winger Ashley Young had a goal disallowed for offside after Van Persie hit the woodwork just before the hour, and City reduced the deficit 60 seconds later through midfielder Yaya Toure before defender Pablo Zabaleta drove in a low shot from an 86th-minute corner.
However, Van Persie had the final say as he netted his 11th league goal since signing from Arsenal for $38 million in August, with City midfielder Samir Nasri guilty of backing away in his position in the defensive wall and sticking out a leg to divert the ball past a helpless Hart in the second of four extra minutes.
City substitute Carlos Tevez, who came on for Mario Balotelli seven minutes after the break, was lucky to avoid punishment after aiming a wild kick at United's Phil Jones late in the match.
Meanwhile, Everton moved up to fourth place with a 2-1 victory at home to Tottenham on Sunday thanks to injury-time goals from Steven Pienaar and Nikica Jelavic.
Clint Dempsey had opened the scoring in the 76th minute for Spurs, who dropped to fifth above West Brom on goal difference.
Liverpool claimed 10th place with a 3-2 victory at West Ham, as James Collins conceded a late own goal after Glen Johnson and Joe Cole scored against their former club.