Yaya Toure and Sergio Aguero scored inside the final eight minutes at Villa Park to seal a 2-0 triumph and close the gap at the Premier League summit to two points ahead of leaders Chelsea hosting Arsenal on Sunday.
The defending champions have already dropped points against Stoke City, Arsenal and Chelsea this term and a determined Villa team, who retained a threat on the counter-attack for most of an absorbing contest, appeared destined to join that list.
But Yaya Toure produced a timely first goal of the season in the 82nd minute before Aguero struck an emphatic second to banish the any hangover from Tuesday night's lacklustre 1-1 draw against Roma in the UEFA Champions League
"I don't know if the word is exactly relief," said Pellegrini in his post-match press conference.
"We deserved to win the game, we had a lot of chances to score.
"I think that we played the best game of this season completely - in possession, in chances to score, in the way we defended, the intensity we recovered the ball [with] and the way we worked trying to score a goal.
"We didn't do it [score] before because Aston Villa defended very well."
Toure has been the focus of criticism over recent weeks, with a pre-season of uncertainty over his future at the Etihad Stadium followed up by a series of below-par performances in the campaign's opening weeks.
But Pellegrini believed the former Barcelona player's attitude and top-level pedigree was always likely to pull him through.
"Of course it is very important for him to score his first goal but Yaya is an experienced player," he added.
"He can be criticised in one or two weeks but I know perfectly the way he thinks and the way he can play.
"I never had a problem about that but it is always important for him to score."
