James Milner opened the scoring seven minutes into the second half before a Vincent Kompany own-goal allowed United back in prior to Sergio Aguero scoring a brilliant winner with just over 10 minutes remaining.
The victory allowed City to go within12 points of league leaders United with seven matches left but Mancini ruled out the possibility of a late-season fightback.
"United? No. The championship is finished," Mancini told Sky Sports 1.
"But we deserved to win because we played better, we played good football. We didn't take any risk.
"This is important for us because we can get second position.
"Maybe we have one game in hand and if we win the game against Wigan we go 10 points in front of Tottenham.
"After that maybe we don't have a problem.
"This is for our supporters because they deserve this victory. After being 15 points behind it was important for them."