Albion struck early to reap the benefits of a positive start as Wales international Jason Koumas skipped past Burnley's midfield combination of Eric Djemba-Djemba and James O'Connor to steer a shot beyond Brian Jensen from 25 yards.
The Baggies supporters were still celebrating that goal when Albion extended their lead as Zoltan Gera crossed from the right and Nathan Ellington headed the ball beyond Jensen with the aid of a deflection off the striker's arm.
Burnley got back into the game thanks to a mistake by Dean Kiely. The Albion keeper quickly threw the ball in the direction of Diomansy Kamara who was beaten by Michael Duff.
The ball then landed at the feet of striker Andy Gray who scored his 12th League goal of the season despite looking suspiciously offside.
Koumas twice went close to extending the visitors lead as he demonstrated his full range of shooting ability from distance.
With half-time approaching referee Jones produced a straight red card for Albion defender Paul McShane after a strong headed challenge on Burnley's James O'Connor which saw the Clarets midfielder leave the pitch on a stretcher, to be replaced by Chris McCann.
Burnley equalised just three minutes into the second half as winger Wade Elliott forced a corner, took it himself to the far post, where Gray had a header blocked before forcing home a shot from close range.
The home team had an opportunity to take the lead as Jon Harley's cross found the head of Elliott but keeper Kiely got down well to his left to make the save.
Kiely then produced an even better save to keep out a header from the dangerous Gray to deny the striker a possible hat-trick.
Burnley's persistence paid off just three minutes from time as full-back Duff crossed for substitute McCann to head beyond Kiely for the winner.
The result leaves Albion with some work still to do to ensure a place in the end-of-season play-offs and the final two games of the season will see nerves tested to their fullest extent at The Hawthorns.