Quinn stepped down as chairman to take on an international development role on Monday, with owner Ellis Short becoming the new chairman with immediate effect.
Saturday's 2-2 home draw with West Brom left the team 16th in the Premier League table and did little to ease the pressure which has been gathering around Bruce in recent weeks, but Quinn believes Short has faith in the Black Cats boss.
"Steve is not isolated by this - it's the exact opposite," the Irishman said in the Daily Mail.
"It's time for him to build a closer relationship with Ellis.
"Steve will get time for this team to gel. There's another window looming. Unless the situation becomes apocalyptic, and it's no good worrying about that, Steve will get time.
"For the overall good of the club this is right. I am happy, absolutely. Ellis is happy - he is speaking to the manager today. You'll find the manager will be happy tonight."
He added: "Ellis will be a great chairman. The beauty of Ellis is that he has a presence. He has authority, he demands serious attention. After three years, he also understands football's snakes and ladders nature, that it's not about under-reaction or over-reaction.
"This can be the making of Steve Bruce. Steve has been in a jumpy world at clubs before, where he read things in the media, but with Ellis he won't. Ellis is straightforward, a real man, no whispering.
"He's not an operator. An operator runs with the hares and hunts with the hounds and to an extent I had to do that.
"Ellis is way above that. Don't expect him to be a media darling. He doesn't worry what's said. And he doesn't need to go out and sell himself because every Sunderland fan knows what he's done in the last three years."
Despite the announcement of Quinn's new role coming in the same statement as the one which revealed he was standing down as chairman, rumours were rife that he was headed for the Manchester City boardroom.
He stressed: "I am not going to Manchester City.
"It surprises me. Everybody knows it's not true yet in the last 24 hours the number of people who have come up to me to say, 'Oh aye, off to Man City', is incredible. I know they know it's not true and they know I know it's not true.
"It's that showbiz thing in football. You have to accept it. It's usually a player going to another club. What I'm meant to say is, 'I'm flattered to be linked'.
"I loved Man City and my time there and I'm delighted for their fans the way things are going - but it ain't my club."