The champions came from behind with goals from David Luiz and Frank Lampard to secure a victory that sees them close the gap between themselves and United to 12 points.
Ancelotti was encouraged by what he saw after a difficult few months for his team.
“It was a good performance and we're still alive,” he told Sky Sports after the game.
“Manchester United are fantastic, they played really well, but we managed to keep going even at 1-0 down, played with power and intensity, so the performance was really good because without playing really well you cannot beat Manchester United.”
The game was not without its contentious moments with Nemanja Vidic sent off late on after picking up two booking whilst David Luiz appeared fortunate to remain on the pitch despite a foul on Rooney, after he had already been booked.
Yet Ancelotti admitted the game had been difficult for the referee and instead explained how his team had managed to change the game.
“In the middle of the first half they were playing very well, they used [Wayne] Rooney and [Javier] Hernandez very well and in that moment we were in difficulty, but we put more pressure in front, we used the midfield, we used pace and in the second half we were very good,” he said.
“There was a lot of intensity on the pitch, so it was not easy for the referee to decide. David Luiz could be lucky, but a lot of things were going on so I think the referee didn’t see."