City overcame the absence of a number of key players to beat Roma 2-0 in the fevered atmosphere of the Stadio Olimpico and claim a place in the knockout stages that seemed unlikely just a fortnight ago.
Pellegrini's men took just two points from their opening four games in the competition but then revived hopes with a stunning win against Bayern Munich, and then similarly stunned the Italians.
Samir Nasri and Pablo Zabaleta struck in the second half as City climbed above both Roma and CSKA Moscow in Group E.
Pellegrini said: "It is not relief because I think we always trusted - especially after we beat Bayern Munich - that we would have a lot of chances to qualify here.
"I am very happy because we won against a very difficult team in a difficult stadium
We played very well from the beginning of the game."
City's last two performances have been in marked contrast to how they began the competition.
It seemed they were heading for a third group stage exit in four seasons after losing to CSKA at home last month and the critics started to come out for Pellegrini.
His team have gone on to defy them but Pellegrini insisted he was not trying to send out any message and immediately turned attention back to domestic matters.
He said: "We recovered and I think we are playing in exactly the same way in the Champions League as the Premier League and you can see that in the last two games against Bayern and Roma.
"We are not trying to send any kind of message to anybody
We are just happy because we qualified for the next stage
That was our target.
"But from tomorrow morning we must forget about the Champions League because we do not have to play until February
We now have four games in December in the Premier League that are so important, more than the Champions League
Every competition is important but I think in England the Premier League is more important than the Champions League.
"We must focus our mind now on trying to beat Leicester on Saturday."
City were without injured captain Vincent Kompany and striker Sergio Aguero while Yaya Toure was suspended
David Silva, on his return from a knee injury, only came off the bench for the final 23 minutes.
Pellegrini said: "Maybe if we had not qualified for the next round it would be easy for me to say we played without all these important players - Kompany, Yaya, Silva and Aguero.
"But I never complain about that
I have a lot of trust in this squad and maybe this squad deserves more credit.
"People say we are just 'Kun' (Aguero)
Kun is a very important player - like Cristiano (Ronaldo) in Madrid or (Lionel) Messi in Barcelona - but I always say this team is not just one player and I think we demonstrated that."
Roma coach Rudi Garcia, whose side now drop into the Europa League, felt the hosts paid the price for failing to make their early dominance count.
Garcia said: "Of course it is about maturity and experience but it is also true we were unlucky.
"They hit the post and the ball went in
When we hit it, it went out.
"Probably we should have made the most of our chances in the first 10-15 minutes
We had a good start and should have scored then.
"We were not effective enough and effectiveness is a trademark of great teams."
Source : PA
Source: PA