Manager Mancini wrote off his side's title hopes after they lost 1-0 at Arsenal to fall eight points behind city rivals Manchester United with only six games remaining, and maintained his side's destiny was out of their hands until they beat United on April 30.
Even then he claimed United remained favourites, but City go into Sunday's final games with a goal difference superior by eight with the teams on level points. They face QPR at home while Sir Alex Ferguson's men travel to Sunderland.
Barry told BBC1`s Football Focus: "We've got an experienced team, an experienced manager, I'm confident we can get the job done.
"He's (Mancini) taken lot of the pressure off the players, that's worked really well for us. We've really raised our game since a couple of disappointing results.
"It looked like we were running away with it at one point, then at one stage bookies were paying out on Manchester United to have won the title."
Victory at the Etihad Stadium on Sunday would secure City's first league title in 44 years, following on from last season's FA Cup success.
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And Barry added: "There's always that weight (of history), we had it last year with the FA Cup.
"Supporters let you know how long it's been and it does build the tension around the place.
"But before the season if anyone had said we'd have a home game against QPR to get three points to win the Premier League title, we'd have snapped their snap hands off to be in this position.
"I'd be lying to say I've not though about (the title) this week, you play all these years to be in a position like this."
But the England international will be taking nothing for granted against a side still requiring a result to ensure their top-flight survival, managed by former City boss Mark Hughes and with City old boys Joey Barton, Nedum Onuoha and Shaun Wright-Phillips among their playing ranks.
"It's going to be a really hard game," he said. "We're playing a QPR side who still need a point for survival and we need one more big effort from the players.
"I've got a lot of respect for Mark Hughes, he had faith in me to sign me three summers ago. I'm hopeful they can stay up but obviously we've got our eyes on the big prize, and that's beating QPR at the weekend.
"I speak to Nedum, when he left he said his worst nightmare would be for QPR to come in struggling for survival and us with a chance to win the league, so I'll leave the texts for after the game."