Favourites City had to do it the hard way as they came from behind to claim the first piece of silverware on offer this season.
Fabio Borini gave Sunderland a first-half lead, but City turned the game on its head with two stunning goals from Yaya Toure and Samir Nasri in the space of a minute early in the second half.
Substitute Jesus Navas ended Sunderland's brave resistance with a third goal late on and Pellegrini felt City deserved their cup success at Wembley.
"I'm very happy," Pellegrini told Sky Sports 1HD. "I think the team deserve this cup because we played well during the whole cup and we scored a lot of goals and conceded just two goals in the whole cup.
"To start winning the first trophy I think is very important for the club, for the players, for me, for the staff - all of us."
Pellegrini admitted his side were far from their best in the first half, but praised their patience in getting back into the game.
"I think that we were not doing things well [in the first half]," noted Pellegrini.
"Sunderland play a very good match, they were very close and we didn't move the ball as fast as we needed [in the first half] but I think in the second half we changed it, we scored two beautiful goals and we had the patience to wait for the moment."
Pellegrini also praised Yaya Toure for hauling his side level with his wonder strike in what proved to be the turning point of the game.
"Yes he is a top player he normally changes the games, but I think also we didn't deserve to lose the first half as it was very close and they score on one counter attack, but second half I think we were the better team."
The win was Chilean Pellegrini's first trophy success in Europe and he hopes it is the first of many at City.
"Very important because at a very important club you can win trophies that for me is the most important thing and we hope it will continue in the future," concluded Pellegrini.