Stoke took the lead from the first chance of the match and it was Fuller who applied the expert finish. John Eustace's pass was flicked on by Mamady Sidibe and Fuller burst clear to place a low shot past the advancing Dean Kiely.
Albion dominated most of the possession and forced Stoke back as Roman Bednar and Zoltan Gera fired shots off-target.
But the visitors struggled to create a clear opportunity as Stoke's occasional raids carried much more potency.
Sidibe and Fuller were proving a real handful and it was no surprise when the latter notched a second seven minutes before the break.
Liam Lawrence's right-wing free-kick found its way to the far post, where Fuller, who had spun off his marker, turned to hammer an unstoppable shot past Kiely.
Albion made a double substitution at the interval, bringing on Kevin Phillips and Ishmael Miller for the ineffective Chris Brunt and Felipe Teixeira.
Wave after wave of attacks were soon raining on the Stoke goal and Albion almost pulled one back in the 63rd minute when Leon Cort cleared Miller's header off the line.
But Stoke were in the comfort zone after Fuller completed his treble with a magnificent individual effort four minutes later.
The Jamaican international picked the ball up on the left flank and tore infield to leave Bostjan Cesar and Pele stranded, before rolling his shot past Kiely.
Albion grabbed a consolation goal 18 minutes from time when Bednar headed in from six yards from Martin Albrechtsen's cross.