City had suffered a 4-1 home defeat by Liverpool last weekend and then a 1-0 loss at Juventus in their penultimate Champions League Group game on Wednesday as their winless run extended to three games.
But Saturday's victory against Southampton, who hadn't lost away from home in the league since the final day of last season, lifted City's spirits and gave Pellegrini the ammunition to silence the critics after Bayern Munich boss Pep Guardiola was once again linked with his job.
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"Yes of course it was a very important reaction. It was not an easy week," said Pellegrini, whose side are above Leicester on goal difference.
"Talking about the Premier League we just won one point from six so it was very important to win against an unbeaten team away. I was happy with the performance.
"I think the game we played against Liverpool was not a normal game for our team or for individual players.
"It is important to have a strong reaction in front of our fans. If you want to be involved in title, you cannot drop more points at home after we already dropped points against West Ham and Liverpool.
"The players gave a very strong reaction. It is not easy because they are playing so many games with the same players.
"We are the only team from the Premier League definitely through in the Champions League and the only team in the Champions League still in the League Cup.
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"It is not a crisis. It is a very good season."
City looked as though they would win even more comfortably when Kevin de Bruyne scored his seventh goal of the season and Fabian Delph his first of the campaign to put the home side 2-0 up after 20 minutes.
But Southampton pulled a goal back through Shane Long's header just after half time and had further chances before Aleksandar Kolarov's goal 20 minutes from the end secured the points for City.
"We played on Wednesday in a very difficult game and we didn't have 72 hours to recover. It was a difficult moment but fortunately the third goal killed the game," Pellegrini added.
City welcomed back David Silva following nine matches out with an ankle injury as a second-half substitute, but will need to check the fitness of Sergio Aguero before their League Cup quarter-final against Championship side Hull on Tuesday.
Pellegrini said he only planned to play the striker for 65 minutes as he continues to be eased back following a hamstring injury, but moments before coming off Aguero appeared to land awkwardly as he challenged at the far post.
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"He has a pain in his heel. We will see tomorrow (Sunday) but I hope it is nothing important," said Pellegrini.
The loss was Southampton's second in a row following a run of eight games without a defeat and manager Ronald Koeman admitted his side had left themselves too much to do following a poor start.
Koeman, whose side play Liverpool in the last eight of the League Cup on Wednesday, said: "If you do that kind of mistakes then you know against City they punish you.
"Maybe we were lucky it was only 2-0 after 25 minutes because we lost every ball. That is not about the systems, it is about playing the ball to the right colour.
"It is important the players take out the positives in the second half. We had the best chance to make it 2-2 with (Dusan) Tadic. But the 3-1 killed the game."
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