City, requiring at least a score-draw to go through to the knock-out phase, were dominated for large periods of the match at a packed Stadio Olimpico where a win would have secured Roma's passage for the first time since the 2010/11 season.
However, Pellegrini's men made up for the absence of key players including top scorer Sergio Aguero and defender Vincent Kompany to turn the match on its head on the hour mark when Samir Nasri fired a superb shot past Morgan De Sanctis and in off the keeper's near post.
It settled City's nerves and after Roma spurned a series of late chances to level the Frenchman coolly set up Pablo Zabaleta inside the box for his first Champions League goal in the 86th minute.
City finished second in Group E behind already-qualified Bayern Munich with Roma finishing third to drop into the Europa League.
But Pellegrini said he had plenty of faith that City, whose Group E campaign was revived by a stunning Aguero treble in a shock 3-2 win over Bayern a fortnight ago, would rise to the challenge.
"Maybe in this moment if we hadn't qualified for the next round, it would be easy for me to say we were missing a lot of players, but I have a lot of trust and confidence in this squad," said Pellegrini.
"I always repeat the same answer. I don't think this team is just one player and I think that today demonstrated that."
City could welcome back Aguero if the Argentinian striker, who has scored 14 Premier League goals and 19 in all competitions so far this season, recovers in time for February's last 16 ties.
Ahead of that challenge, Pellegrini was quick to play down City's dreams of making it into the quarter-finals of the competition for the first time in the history of the competition.
"We're not trying to send any message to anybody. We're just happy to make it to the next stage, which was our target from the start," said Pellegrini, whose side were eliminated in the last 16 by Barcelona last year.
"Last year we qualified with 15 points from 18, the best second in the history of the champions league, then we played against Barcelona and you can't afford to make the kind of mistakes we made against them."
As City look ahead to February, Roma turn their attention back to Serie A and the Europa League.
For Garcia, it was all part of a "learning experience" for the club which he has helped turn into title contenders for the first time in the best part of a decade.
But he admitted the hosts should have been more clinical early on.
"We just didn't capitalise on our chances, especially in the opening 15 minutes," said Garcia. "We need maturity and experience but we were also unlucky today. (Kostas) Manolas hit the post and we missed other opportunities."
He added: "Efficiency is a trademark of great teams and they were a lot more efficient than us in that respect tonight.
"City didn't start their group campaign very well but improved as time went on. Even with so many players missing they still have a top quality side, and part of that is down to the budget they have.
"It was avery tough group, especially with Bayern in there. Going through would have been a feat and upset all odds."