City started that quest with a 3-1 victory over Sunderland at Wembley on Sunday, but had to come from a goal behind after Fabio Borini's first-half opener.
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The response after the interval was emphatic, with Yaya Toure levelling from distance before two-time Wembley loser Samir Nasri's first-time shot put City ahead a minute later.
Jesus Navas made the closing stages more comfortable with the third to leave Pellegrini delighted, though he appeared to acknowledge that more glory must follow if he is to maintain the faith of his employers.
"I'm very happy," he told Sky Sports. "The team deserve this cup because we played well during the whole cup and scored a lot of goals.
"It's important for all of us - for the players, the club, for me, for the staff - all of us. I'm at a very important club, where you must win trophies."
Pellegrini was unhappy with City's first-half showing, but also paid tribute to Sunderland for making life difficult.
"Sunderland played a good match," he continued. "We didn't move the ball as fast as we needed (in the first half) but in the second half we scored two beautiful goals and had the patience to wait for the (right) moment."
Victory for City eradicates memories of the disappointing 1-0 FA Cup final defeat to Wigan Athletic in May and means hopes of a quadruple remain on course.
City trail Chelsea by six points in the race for the Premier League title, though that gap will be cut if they win both of their games in hand over Jose Mourinho's men.
A chance to avenge the FA Cup final loss awaits when City meet Wigan in the quarter-finals of this year's competition, and they are still in the Champions League, though Barcelona will take a 2-0 lead into the second leg of their last-16 tie.