City, who were beaten by Juventus in the Champions League and West Ham in the Barclays Premier League in the space of five days last week, raced into a 4-0 lead inside 36 minutes at the Stadium of Light on their way to a 4-1 third-round win.
It was the perfect response for Pellegrini, who fielded a strong side despite missing eight senior men through injury.
He said: "Really it was important to win the way we did today because we have just come from two important defeats, one in the Champions League and the other in the Premier League, so it was important to start this Capital One Cup in the way that we did in the first half.
"It's important to win
Maybe we didn't deserve to lose the two games that we lost, but we did, so it was important not to continue in that sense."
Sunderland's fate was decided inside 26 catastrophic first-half minutes as a Sergio Aguero penalty, Kevin de Bruyne's strike, a Vito Mannone own goal and Raheem Sterling's 36th-minute effort gave the visitors an unassailable lead.
The home side felt Patrick van Aanholt's challenge on Jesus Navas which led to the spot-kick might have been outside the area and were aggrieved to see a Fabio Borini equaliser chalked off for a foul on Nicolas Otamendi, but ultimately Ola Toivonen's 83rd-minute consolation header was all they had to show for their efforts.
Aguero, De Bruyne and Sterling, who cost a combined total of around £140million, were simply irresistible, although Pellegrini was unwilling to label them the best front three in the country.
He said: "We will see
We have a lot of games to play in the league
They are very good players and at the end of the season, we will see which are the best players in the league.
"It's important now to be focusing game-by-game and not to think about individual players."
Opposite number Dick Advocaat, whose side sit at the foot of the Premier League table, was dismayed by the result against the leaders, but buoyed by elements of the performance.
The Dutchman said: "Maybe you think I am negative, but I am negative, but I am not
Yes, negative that we lost 4-1, but the way we played in the first 25 minutes, it was very positive.
"I was not really happy with the way that we again gave the goals away, and it's not the first time, but there's no sense in calling names because everybody could see the mistakes, simple mistakes and it's not possible [to make those mistakes] at this level, especially not against a team like that.
"But before that, when they scored, 1-0, we had a good chance to score, 1-1
At the moment, everything is not rolling in our way
But at a certain moment, that will change.
"I said at half-time to the players, 'Now you have to show commitment for the team, but also as an individual because if you go on the same as we did in the last 15 minutes, then they will score four goals more'.
"But I was very pleased with what I saw in the second half
They worked very hard
You could see from the fans - they stayed until the end
But the quality from the other side was too much."
City went into the game without David Silva, Samir Nasri, Fabian Delph, Pablo Zabaleta, Vincent Kompany, Eliaquim Mangala, Gael Clichy, Wilfried Bony and Kelechi Iheanacho, but Pellegrini is hopeful of having at least a couple of them back for Saturday's league trip to Tottenham.
Source : PA
Source: PA