Albion have now failed to pick up a single point since January 20 and are 10 adrift at the bottom of the table, seemingly resigned to Championship football next season.
Burnley, previously winless in five Premier League encounters with West Brom, started brightly and came close to an early goal when Barnes blazed wide.
The Clarets edged in front 23 minutes in, though, thanks to an acrobatic volley from Barnes at the end of Aaron Lennon's cross, leaving Foster with little chance between the sticks.
Unbeaten in their previous 13 games in which they have gone ahead, the visitors pushed on for a second and nearly got it through a Wood header towards the far post against his former club.
Wood and Rondon then tried their luck with attempts on target, both of which were kept out as Burnley took a slender lead into the interval.
A decent opening was passed up by Jay Rodriguez as Albion attempted to draw themselves level, as he failed to keep his header on target.
Ashley Westwood drilled wide before Barnes did likewise, but the visitors' second arrived 17 minutes from time after Wood reacted quickest to head the ball over the line following an initial save from Foster.
Rondon did restore a little hope by lashing the ball past Pope after James Tarkowski failed to clear it, although Burnley felt that the goal should have been ruled out as Stephen Ward was injured on the ground.
West Brom's late chance to snatch a point could not be taken as Chris Brunt failed to adjust his body when the ball fell to him, meaning a third win in succession for Europe-chasing Burnley.
WEST BROMWICH ALBION (3-5-2): Foster; Dawson, Evans, Hegazy; Nyom, Livermore, Yacob, Brunt, Gibbs (Phillips 46'); Rodriguez (Burke 74'), Rondon
BURNLEY (4-4-2): Pope; Lowton, Ward, Tarkowski, Mee; Westwood, Cork, Lennon, N'Koudou (Hendrick 69'); Barnes, Wood (Vokes 77')