The Baggies cruised through a totally dominant first half at a drizzly Gigg Lane.
The Championship side went in front after just five minutes. Filipe Teixeira robbed Bury skipper Paul Scott just outside the hosts' box and squared for Graham Dorrans who coolly slotted home from 12 yards.
Simon Cox saw an effort ruled out for offside before Albion doubled their lead after 26 minutes. This time Craig Beattie crossed from the right and Bury defender Michael Jones could only divert a low ball into his own net from close range.
The League Two side came close when Ryan Lowe was thwarted by a brave save from the experienced Dean Kiely and then after 68 minutes Efe Sodje rose to head home only to have been ruled offside.
Lowe drilled a low shot inches wide late on as the visitors held on for a comfortable victory.