Pellegrini and his players had been heavily criticised in recent weeks for a run of six defeats in their previous eight games that included a 4-2 reverse to United at Old Trafford last weekend and has seen them drop out of contention for the title.
But City produced a much improved performance at the Etihad Stadium on Sunday to see off West Ham with an 18th-minute James Collins own goal and Sergio Aguero's effort in the 36th minute.
The victory was however soured by a facial injury to David Silva, suffered after 67 minutes when he was caught by an elbow from Cheikhou Kouyate.
"It was very important for our team to win because we had come from two defeats in a row and when one of those defeats is a derby it's worse," Pellegrini said.
"It was very important to try to be a consistent team and I think that we played - especially the first 65 minutes until the injury to David - as a consistent team."
"I think that we played a whole game."
