But, just two years after joining Albion in 1977, the winger made the historic £1million move to become the first English player to sign for Real Madrid.
The former Arsenal schoolboy joined the Baggies in March 1977, signed by Johnny Giles, around his 21st birthday and made just 86 appearances but more than made his mark on the club, the First Division and the continent in his two years.
Cunningham had already become the first black player to represent England under-21s but under the stewardship of the attack-minded Ron Atkinson, a lover of flair, he glistened and during Albion's stunning form both home and abroad, he caught the eye of superpowers.
Indeed, Spanish scouts were present to watch Albion tackle Valencia in the UEFA Cup at the Mestalla on November 22 1978. The legend goes they were there to watch the great Argentine Mario Kempes, but after a 1-1 draw in which Cunningham equalised, it was the Albion No.11 they left talking about.
The Madridistas took note and shelled out £950,000 to take him to the capital. In the February of 1980, his first season in Spain, Cunningham became one of very few Real players ever to earn a standing ovation at the Nou Camp, home of arch-rivals Barcelona, after a dazzling display in a 2-0 win.
Papers headlined it 'The man who ran riot in the Nou Camp'. One onlooker was quoted: "It was like seeing (Johan) Cruyff but with black skin."
Cyrille Regis, Laurie Cunningham, and Brendon Batson (AMA)
Before that notoriety for carefree Laurie it was The Hawthorns, where there was a brotherhood with fellow Londoners Cyrille Regis - who moved as a child from French Guyana - and likewise Brendon Batson, who did so from Grenada. As a trio they would become The Three Degrees.
"Cunningham was ahead of his time. At the end of the day he'd take ballet lessons at Orient to improve his balance," says chairman of Albion supporters club John Homer, who was in the stands to watch the winger and Co. destroy most who came before them.
"He was a graceful, balanced player. There was that famous Ron Atkinson quote that Laurie could walk on snow and not leave an imprint. That says a lot."
Indeed, the north London-born winger would pay for £1 fines for being late during his time at Orient by winning dance competitions.
It was that balletic poise and grace that fans and colleagues recall so well.
"I was fortunate to see his debut at Spurs, which was March 1977," Homer recalls. "He made his home debut on March 16, we thrashed Championship-chasing Ipswich 4-0, Laurie scored and Bryan Robson got a hat-trick. It was my 22nd birthday and a very happy present.