City are now top of the Premier League on goal difference with two games to play.
"It's a great victory but we have two difficult games left and United have two easy games," said Mancini.
Phil McNultyChief football writer, BBC Sport
Andre Marriner's final whistle was drowned out by the deafening delight of Manchester City supporters celebrating victory in the biggest game in 20 years of the Premier League. It was the sound of Ferguson's worst nightmare moving closer to reality - namely United being denied a 20th title by the rivals he has kept down at heel for pretty much all of his 25 years-plus at Old Trafford.
However, United boss Sir Alex Ferguson insists City are "in the driving seat" to clinch the title.
Vincent Kompany scored the winner just before half-time to send City top.
Their goal difference is now eight better than United's and if they win their last two games at Newcastle United and at home to QPR on the final day of the season they will almost certainly be champions.
Mancini said: "We now have the chance for the title in our hands. Manchester United are still favourites. Swansea and Sunderland [United's opponents] have nothing to play for. I am very happy that we are top but it doesn't change anything. Football is crazy.
"We were on the top for 28 games and I think we deserve another chance. But it is not finished. If we think it is finished, we will be making a big mistake."
Ferguson admitted it was a damaging result.
"They are in the driving seat and we are up against it," he said. "They only need to win two more games and they win the league."
United were eight points clear but lost 1-0 at Wigan and threw away a 4-2 lead over Everton to draw 4-4 at Old Trafford.
"You can look back at all sorts of games but you've got to look at the Everton one," said Ferguson.
"That was a ridiculous performance for the last seven minutes to lose two goals. That's maybe cost us the league."
As United have been faltering, City have been registering impressive victories at home to West Brom (4-0) and away against Norwich (6-1) and Wolves (2-0).
'No complaints' from Ferguson
Ferguson admitted his side had not been at their best on Monday and that City deserved their win, given to them by Kompany's powerful header from a David Silva corner in first-half injury time.
"We didn't test their goalkeeper enough," he said. "We started quite brightly and for 15 minutes our possession of the ball was good, but the longer the half went on I thought we were looking for half-time.
"I'm very annoyed to concede from a set-piece. At this level of football you can't do that.
"It was a bad time to lose the goal. After that they [City] just sat and played on the counter-attack. But I can't complain about the result."
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