The Baggies landed the 26-year-old on a free transfer last week after he opted to run down his contract at Reading.
A number 10 who can also play as an attacking central midfielder, Swift scored 11 goals and registered 13 assists for the Royals last season.
And Quashie believes Albion were badly lacking a player like Swift last year.
“One thing that stands out for me in the Championship, if you want to get promoted you need to play with a good, attacking, number 10,” Quashie said.
“We had (Matheus) Pereira when we went up last. You look at Aston Villa when they went up. You need that creative player in the middle of the park.
“You need that number 10/number eight that can really contribute goals and assists but I don’t think Albion had that last year.
“I hope John Swift comes in and does really well. He has come from a good upbringing. He was at Chelsea as youngster. He has gone on to Reading and played really well.
“Every time I’ve seen him I’ve been really impressed with him. His energy levels are good. He gets around the park well.
“And I just think with that stability in the middle of the park it will give him the licence to go on and create a little bit more.”
Albion finished 10th in the Championship last season – their lowest league finish for 20 years.
But Quashie believes finishing in the top six has to be the minimum target for next year.
“I would say the top six as we stand but I still think there should be more than enough to push on,” he said when asked about Albion’s targets next season.
“I don’t know if the top two will be a certainty when you look at the teams coming down and the parachute payments they will have.
“That will play a massive part I think. But they have more than enough to be involved in the top six. And once you are in that top six you can start looking up.
“They are good enough to do that – 100 per cent. We believe in them as ex-players. The supporters believe in them and people I speak to at the club believe in them.
“But it’s up to the players to believe in themselves a bit more and give everyone that feeling they can really have a go at it.”