Pellegrini's City won the Capital One Cup on Sunday afternoon with a thrilling 3-1 win over Sunderland at Wembley.
City looked destined for their second shock cup final defeat inside 12 months when Fabio Borini gave the Black Cats a deserved first-half lead.
But Yaya Toure and Samir Nasri scored within two minutes of each other to put City ahead and Jesus Navas ended any hopes of an upset in the final minute.
City are still in the FA Cup and they lie second in the Barclays Premier League. They are two goals down in their Champions League tie with Barcelona, but Pellegrini hopes he can guide his team to glory in all three remaining competitions.
"We are the only club that has the chance to continue trying to win all the competitions and this gives us a lot of confidence," the City manager said.
"We wanted this trophy but I don't think that anybody can think that is enough. I don't think for top players or important clubs, you are satisfied with one trophy.
"We know it won't be easy, but we are going to try."
There was no way City looked capable of launching such a devastating comeback at the end of the first half.
Sunderland ran the show in the first 45 minutes and when Borini shrugged off Vincent Kompany to fire past Costel Pantilimon from a tight angle it came as no surprise.
The favourites would have been 2-0 down had Kompany not denied the Italian with a superb tackle just before the break.
In the end it took two "wonder-strikes", as Gus Poyet called them, to put City in front.
Toure curled a dipping shot over Vito Mannone from 30 yards to level and Nasri clipped a sweet half-volley just inside the Sunderland stopper's far post from Aleksander Kolarov's pass.
Sunderland manager Poyet was stunned by the quality of both goals.
"I am proud of my team," Poyet said.
"For 45 minutes we were more than decent but then Toure scores from 35 yards in the top corner. Maybe if we had two goalkeepers we would have stood a chance."
Poyet was happy with his team's overall performance, but he could not hide his disappointment at losing a cup final.
"It's not nice when you are on the losing side," the Uruguayan said.
"I hate it but someone needs to win. It's a final.
"The difference was two special goals. We didn't take our opportunities and then it was just like 'bang-bang' and it's over.
"We tried our best. Now we just need to try to stay in the Barclays Premier League."
Poyet feels it could have been a different story had Martin Atkinson stopped play in the build-up to Toure's strike, though.
"On another day we would have got a foul on Seb Larsson in the run-up to the goal," he said.
"If someone puts the studs into your foot when you clear the ball it's a foul.
"But I am not going to be a hypocrite like most to blame the referee because Yaya Toure is an outstanding player, but it was a foul.
"It was a very easy foul to give. Maybe you can call the ref and ask him, he will have an excuse. They always have."
Both Poyet and Pellegrini will now ready their teams for FA Cup action next week.
Sunderland play Hull while City face the team that beat them in the final last year - Wigan.
Pellegrini reckons the memory of last year's FA Cup final defeat was one of the reasons why his team under-performed in the first half.
"Maybe what happened last year in the final was on their minds in the first half but I think that the most important thing was to calm the players and trust in them," Pellegrini said.
"I told the players they had 45 minutes to change the score. It was important for the players to be calm and they were."