Burnley's task looked to have become easier when West Brom's Semi Ajayi was sent off on the half-hour mark, his handball deemed to have denied the hosts a goalscoring opportunity.
Despite enjoying plenty of possession, however, Burnley created very little in attack, disappointingly mustering just one shot on target all match.
The depleted visitors had two great chances to win it late on, but January signing Mbaye Diagne was unable to find the target from close range and Matheus Pereira's goalbound strike was cleared off the line by James Tarkowski.
Even though they failed to secure all three points, Burnley stretched their unbeaten run to four league games to sit 15th in the standings on 28 points, nine points clear of the relegation zone.
West Brom, however, will rue those missed chances as they remain in 19th on 14 points, 11 from safety having won just two of their 25 league games this season.
"The quality of the performance here today shows we have come a long way in a short period of time," West Brom boss Sam Allardyce told Sky Sports.
"The sad thing is it hasn't bought us three points. We have to start doing the final side of performances which is putting the ball in the back of the net and winning games."
Despite the visitors having a numerical disadvantage, Burnley laboured and ultimately were not deserving of the victory.
"It's the worst I've seen us in a while," Burnley defender Tarkowski told Sky Sports. "You'd think we were the ones with 10 men. We were not as organised as we should be and we had to stick in the game. The only positive is that we didn't get beat."