And Quashie has backed the experienced manager to get more out of a Baggies squad that massively underperformed last year.
Bruce replaced the sacked Valerien Ismael as Albion boss in February last year following a dismal run of just one win in 13 games.
But the 61-year-old didn't have the impact he expected when arriving at The Hawthorns with the team going to finish 10th in the Championship.
Bruce, though, has won promotion into the Premier League four times before in his career. And it's that record that Quashie feels makes him the right man to take Albion forward.
"He (Bruce) is known for getting teams promoted so at this moment in time I think it’s a good fit," the former Scotland international said.
"I think they got better after he came in and now it's just trying to find that right balance in the squad.
"Are players staying, going? What is the infrastructure? How does he see it and how do the board see it?
"How they recruit is going to be so important. It’s a tricky time but get it right and it’s a club that can really push on.
"But I know Stephen Clemence very well. He is a great coach. He has been very successful with Steve in terms of getting the best out of players.
"And what I will do is wish them the best of luck because they have shown they can get the best out of some clubs and that is hopefully something West Brom will get the benefits of and the rewards."
Quashie was disappointed with the way Albion's players performed last season in the final stages of Ismael's tenure – with the Frenchman initially starting well with a 10-game unbeaten run.
"I want the club to do well and I wanted him (Ismael) to come in and hit the ground running which I think he did," Quashie added.
"I think the style of football, I mentioned a few times, that it was going to take it’s toll on the players because of how intense it was.
"And then they got a few injuries, they started to creep in and then there were a few games where they weren’t playing very well and things started to leak out of the changing room and into the press.
"People falling out with managers. You look at it and think that shouldn’t be happening.
"From what I was reading, you could tell there was unrest.
"But one thing you still have to do as a player is give 110 per cent.
"If you don’t get on with a manager you still have to do it for the supporters of the football club and you still have to do it for yourself and for your family."