By Barnes’ overhead volley
Pope plucked the ball from beneath the crossbar.
However, Burnley took the lead midway through the first half as Barnes scored for the fourth successive game.
The ball fortuitously ricocheted into the path of Aaron Lennon down the right.
His cross was slightly behind Barnes, but the striker hooked a volley over his shoulder and the ball flew past Ben Foster and into the top corner of the net.
West Brom had it all to do now. The Baggies had not come from behind to win a Premier League game since February last year.
As they looked to get back into the match, Alan Pardew’s team were taking chances at the back and leaving themselves open to the counter attack.
Georges-kevin Nkoudou caused them numerous problems with his pace and his cross-field pass should have brought a second goal, but Wood dragged his shot wide from inside the penalty area.
West Brom enjoyed their best spell just before half time, when Rondon brought a decent low save from Pope with a glancing header from Jake Livermore’s cross, before Rodriguez sliced a shot wide.
Pardew made an attacking substitution at the break, bringing on winger Matt Phillips in place of full-back Kieran Gibbs. His side saw much more of the ball during the early stages of the second half and Rodriguez twice nearly got on the end of dangerous crosses into the penalty area.
However, the all-important second goal of the game went Burnley’s way, when Wood headed the ball over the line from close range after Foster had blocked his initial attempt.
That sparked a mass exodus of home supporters, but those who left missed Rondon’s reply. The Baggies striker neatly nicked the ball away from James Tarkowski inside the penalty area, before firing a low shot into the bottom corner. It proved to be in vain, though.