Tony Pulis' team played with great commitment and desire, buoyed as they were by their second-minute goal, and are now six points clear of the bottom three.
But West Brom's hopes of surviving relegation fade with every game and Tony Mowbray's team remain rooted to the bottom of the table with 24 points.
There is no faulting the ambition of the Baggies or their commitment to play attractive football but they needed a positive result against a team who had previously mustered just four points from a possible 45 on their travels.
Mowbray admits to West Brom failings
Baggies manager Mowbray and his players had made positive noises during the week about avoiding the drop but their early defending suggested a team heading straight back to the Championship.
Martis failed to deal with a bouncing ball and was dispossessed by Fuller, who closed in on goal before unleashing a low strike that Carson allowed to creep under his body.
It was a passionate and frenetic opening to the game - and the Baggies goalkeeper was soon back in the thick of the action.
He partially atoned for his failure to stop Fuller's strike by brilliantly tipping a wonderfully flighted free-kick from Glenn Whelan round the post.
The home team appealed for a penalty after Ryan Shawcross appeared to impede Marc-Antoine Fortune but referee Martin Atkinson was unimpressed and in any case the initial contact was made outside the area.
However, James Morrison should have equalised for the home team when the ball dropped invitingly for him 14 yards from goal but the unmarked midfielder did not connect cleanly and Thomas Sorensen saved.
Jay Simpson shot narrowly wide from 25 yards, Paul Robinson wasted an opening with a heavy touch and three West Brom players failed to connect with a deliciously floated Chris Bunt cross as the Baggies responded to the impassioned pleading of their fans.
But it was Stoke who finished the opening half on top, with a strike from Whelan deflecting wide, and for all West Brom's endeavour a lack of precision at the heart of their play ensured they went into the break trailing.
Early in the second half the ball broke invitingly to Beattie after two Baggies' defenders failed to clear the danger and the striker made a sensational contact with his left foot.
We still have a lot to do - Pulis
Carson got a hand to Beattie's 14-yard strike but could not prevent Stoke from doubling their advantage.
Stung by the second goal, the Baggies went on the attack and Mowbray made a bold triple substitution, introducing Robert Koren, Filipe Teixeira and Roman Bednar.
Brunt struck the crossbar with a thunderous swerving strike from almost 30 yards and Shawcross headed narrowly over his own goal from a Koren cross.
But Stoke, well-organised and physically imposing, held their shape and maintained their discipline.
West Brom did put the ball in the net when a long-range strike deflected off Fortune but the goal was ruled out for offside.
West Brom boss Tony Mowbray:
"It leaves us with a mountain to climb, but as I said to the players in the dressing room we have to keep going.
"It was hugely disappointing. We fell short at both ends of the pitch.
"We do concede some very poor goals and we were not good enough to keep them out. The concentration levels were not good enough."
Stoke boss Tony Pulis:
"It is a monkey off our back. It was a very important game but they all are now.
"We have played well away from home before but not had the results. It was a smashing result but we move on now.
"We have a lot of work to do. The important thing to do is to keep our feet on the ground."
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