A goal on the stroke of half-time by winger Wade Elliott and a second-half double from striker Steven Thompson were enough to send the Clarets through to the fifth round.
Robbie Blake gave an outstanding performance as he constantly tormented the Albion defence and was unlucky not to get on the scoresheet himself on a number of occasions.
A free-kick from Blake after 28 minutes flew inches over the crossbar while three minutes later the same player collected a pass from Chris McCann but placed his shot just wide of the post.
Former Baggies goalkeeper Brian Jensen did well to tip over a shot from Roman Bednar and produced an even better save to keep out a header from central defender Ryan Donk.
Burnley took the lead in stoppage time at the end of the first half when a free-kick from Jensen midway inside his own half wasn't dealt with by the Albion defence and Elliott volleyed over Scott Carson from 20 yards.
The visitors should have equalised in the opening minute of the second half but Bednar glanced his header wide.
That miss was to prove costly as Christian Kalvenes crossed the ball from the byline on the left where Martin Paterson's knockdown was converted by Thompson.
Albion got back in the game when Gianni Zulverloon seized on a mistake by Michael Duff and found the far corner of the net with a right-foot shot.
Blake had a great chance to increase Burnley's lead but his left-footed effort lacked the power to beat Carson.
The Albion goalkeeper then fumbled a free-kick from Blake and was fortunate that Paterson didn't convert the rebound.
The tie was secured two minutes from time after magnificent approach play from Blake who twisted and turned his way to the byline before setting up Thompson to convert from close range.