The Italian boss repeatedly insisted that Manchester United were favourites in the title race, even after Vincent Kompany's first-half header against the Red Devils put City on the top of the table at the end of April.
Barry revealed that Mancini showed his experience of title run-ins; his mind games eased the tension in the City camp and the team's performances improved once United began being tipped for the title.
"We've got an experienced team, an experienced manager, I'm confident we can get the job done," Barry told The Telegraph.
|MANCINI NEVER LOST FAITH
|Manchester City manager always believed his team would get back in the title race
"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 against QPR on Sunday will secure a first league title for the blue half of Manchester in 44 years and Barry says the week has been dominated by the thought of lifting the title and he can feel the weight of tension amongst the City fans.
"Supporters let you know how long it's been and it does build the tension around the place," he added.
"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 thought about [the title] this week, you play all these years to be in a position like this."
On Sunday the former Aston Villa star will come up against Mark Hughes, the manager who signed him for City. Barry hopes QPR can stay up but not at the cost of City's Premier League triumph.
|10/1||Manchester City are 10/1 to beat QPR 3-1 with bet365|
"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."