Carlos Corberan's side has now failed to score in their last four home games and did not register a shot on target in the first half on Thursday, while Burnley found it equally as hard in front of goal.
However, West Brom had Alex Palmer to thank when he stopped Jeremy Sarmiento one-on-one.
Burnley seemed to want the win more and had a goal chalked off in rather controversial fashion as Jaidon Anthony's seemingly perfect header was belatedly ruled out by referee Tom Nield before Lucas Pires found the goal frame.
Neither side was able to find the back of the net again which meant their respective winless and scoreless runs continued.
The hosts posed the first threat of the contest when Darnell Furlong swept a deep cross towards the back post and an unmarked John Swift, who volleyed over the crossbar.
Burnley were the brighter of the two in the opening quarter and began to put more pressure on the hosts' goalmouth, Pires blasting wide from distance with another early warning shot.
The away side should have opened the scoring before the half-hour mark as Swift's poor backpass put Sarmiento through on goal but he was heroically denied by a scrambling Palmer.
The Baggies rode the storm and started to grow into the contest themselves. Callum Styles gave himself space and let fly from 20 yards with an effort that flew marginally over the upright.
The Clarets registered the first shot on target of the half as Zian Flemming decided to have a go from distance which rolled straight into the grasp of Palmer.
Burnley were forcing the issue more than their lacklustre opponents; Josh Cullen headed into the path of Sarmiento, who aimed straight at Palmer once again.
The visitors thought they broke the deadlock with an hour gone.
Pires picked out Anthony to nod into the back of the net but referee Nield cancelled the goal out for what appeared to be a soft push on Styles which left Burnley boss Scott Parker scratching his head on the touchline.
The Baggies were struggling to deal with Pires and this time Palmer could only watch his effort roll onto a post and away from danger to keep it scoreless.
It was evident to see West Brom's lack of confidence in front of goal and it took until the 65th minute for them to have their first shot on target - a harmless looping header by Karlan Ahearne-Grant was easily collected by James Trafford.
Baggies fans lacked trust in their toothless attack and many headed for the exit doors in the 85th minute before boos rang around The Hawthorns on the final whistle, as Burnley extended their scoreless run to three.
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