West Brom were next to come close via Marc-Antoine Fortune after six minutes, but he fired wide from inside the area.
Peter Odemwingie twice came close for the home side after 12 minutes, but the Nigerian fired wide from close range.
West Brom continued to enjoy the better of the opening period as the match approached the 20 minute mark, but R's skipper Joey Barton had an effort blocked by Chris Brunt after their best move of the match.
West Brom took the lead, however, after 22 minutes when Dorrans found the net with a powerful effort from distance, albeit with a helping hand from QPR stopper Paddy Kenny.
It was West Brom who started the second period the better and they so nearly moved 2-0 ahead after 55 minutes, but Billy Jones rattled the post from inside the box.
Barton had QPR's first real chance of the second period after 60 minutes, but he struck wide from the edge of the box.
Bobby Zamora had a glorious chance to level after smart work from Taarabt after 76 minutes, but his effort struck his own man Jamie Mackie from close range.
The result leaves West Brom in 13th spot in the table, while QPR sit in 16th.