The Baggies landed Swift on a free transfer last week after the 26-year-old opted to run down his contract at Championship rivals Reading.
Widely regarded as one of the best players in the division, the attacking midfielder scored 11 goals and registered 13 assists for the Royals last season.
He joins an Albion side who badly lacked creativity last year with manager Steve Bruce not having a natural number 10 in his squad that he could call on.
And Robertson – who made more than 600 appearances for the club between 1969 and 1986 – is expecting the former Brentford man to make a big difference next season.
“John Swift is exactly the type of player we needed to sign,” the Baggies legend said.
“We needed to bring someone in who can make something happen.
“If we’re being honest, we haven’t had that type of player ever since Matheus Pereira left.
“I was actually surprised when I saw the Swift deal was done.
“He’s been linked with a lot of clubs and I thought he might be tempted by a move to the Premier League.
“But I’m absolutely delighted he has decided to join us.
“When you look at his history and the season he had last year, the number of goals and assists he got for a struggling Reading side is very impressive.
“And I think it’s great we’ve managed to get him in so early.
“His arrival has given everyone a lift and it shows the fans Bruce knows exactly what we need and that he can get these deals done.”
While pleased with the addition of Swift, Robertson is hoping a few more players arrive in the coming weeks with Jed Wallace known to be one player the Baggies are eager to sign.
“I think the arrival of Swift just goes to show that Albion are still a very big draw,” Robertson continued.
“Yes, we had a poor season last year. It wasn’t good enough.
“We had a terrible run and we saw Valerien Ismael sacked as manager.
“But with the right additions we have to be aiming to be compete at the top again next season.
“Swift is the first piece of the puzzle.
“And now we need to make three or four more additions to make sure we get to the top.
“I think Bruce needs to be looking at making three or four more permanent additions.
“And then I think it’s important he tops up the squad with some quality loans from the Premier League.
“Those players don’t tend to arrive until later in the window.
“But Bruce knows everyone in football. He’s got great contracts. So hopefully he’ll get a couple of exciting loanees as well.”