The hosts threatened early on as the woodwork prevented Steven N'Zonzi from steering the ball into his own net, before Stoke began to dominate and Ben Foster denied Jon Walters with a smart reflex save.
Walters was then guilty of wasting a good chance when he headed Peter Odemwingie's cross wide from close range, but Mark Hughes' side deservedly took the lead after 22 minutes.
Marko Arnautovic strode into the area and his low cross was diverted beyond Foster by the legs of the unassuming Gareth McAuley as he chased back into the box.
The Baggies showed signs of improvement after the break and found themselves level through Stephane Sessegnon, who calmly guided Craig Dawson's low cross beyond Thomas Sorensen.
The woodwork denied second-half goals at both ends as Arnautovic and Youssouf Mulumbu both clipped the post, before Adam stepped off the bench to net the decisive goal with a fine low strike from the edge of the box.
For all the irrelevance of the game barring the impossibility of a 17-goal swing involving doomed Norwich at Carrow Road, both sides - and their supporters - had done their best to make it a memorable one.
Stoke might have been missing winger Oussama Assaidi due to a knee injury but they controlled the first half with a brand of slick, passing football few could have imagined at the same stage last season.
The Baggies had the first chance when Graham Dorrans' corner from the right was headed down by Dawson and Sessegnon pounced at close range only to see his effort ricochet off N'Zonzi and strike the underside of the bar.
James Morrison almost found Saido Berahino in front of goal then Sessegnon had a shot deflected - but after a frantic first few minutes the visitors seized control of the game.
Walters flashed a shot across the face of goal which was brilliantly saved by Foster, then Geoff Cameron dug out a deep cross from the right which Walters headed inches wide.
Stoke grabbed the lead in fortuitous fashion in the 22nd minute when N'Zonzi fed Arnautovic on the left and his attempted cross hit McAuley and deflected into the net past a helpless Foster.
If there was a degree of luck about the visitors' opener it was more than deserved on the basis of their dominance of play in the first half-hour.
A frustrated McAuley was given a stern ticking-off by referee Lee Probert for hacking down Peter Odemwingie on the halfway line, then Morrison did his best to drag the Baggies out of their end-of-season torpor with a shot that strung Stoke goalkeeper Sorensen's hands on the stroke of half-time.
Mel's men made a brighter start to the second period but they were almost hampered by the exploits of Foster, who came close to hurling the ball into his own net after gathering from Odemwingie on 50 minutes.
The Baggies grabbed the equaliser they just about deserved in the 56th minute when Dawson raced into space on the right and sent in a cross which Sessegnon met with a sweet shot past Sorensen and into the net.
Arnautovic responded almost immediately with a shot from the Stoke left which flashed inches past Foster's left-hand post, but the Baggies largely retained control and could have gone ahead with two chances in as many minutes.
First Sessegnon headed a Dawson shot straight at Sorensen then Mulumbu worked into space on the right side of the Stoke box and unleashed a fierce drive which crashed back off Sorensen's right upright.
Baggies substitute Victor Anichebe blazed over the bar before Adam drifted across the retreating West Brom rearguard before hitting a low shot across Foster and into the corner of the net for the winner.