In a closely-contested match it was Rovers who got their noses in front just before the half hour when Josh Morris' free-kick was nodded into the back of his own net by Albion defender Donervorn Daniels.
The home side came back into the game and they were unlucky not to level when Romaine Sawyers saw his shot across goal cannon off the far post.
After looking solid in the first half Rovers were undone seconds after the restart when West Brom levelled. Failure to clear the ball in the centre of midfield proved costly and a cross from the left was bundled home, the last touch possibly coming off forward Saido Berahino.
Suddenly West Brom were looking the livelier of the two sides, Berahino flashed a shot narrowly wide, before Rovers were grateful to goalkeeper Seb Usai as the Albion striker was denied once more.
It proved to be a crucial save as on the hour Rovers burst away with Micah Evans and although Jamie Maclaren couldn't reach his centre the ball eventually came out to Hugo Fernandez and he clipped the ball into the top corner of the goal.
The home side came back and again some hesitant defending proved costly as Sawyers fired low past Usai to level nine minutes from time.
As the game entered the final stages it was West Brom who looked the more likely to grab a winner as they enjoyed plenty of possession but Rovers held on to claim their first point of the season.
"It was a competitive game and I was pleased with the way our lads went about it," said coach Gary Bowyer.
"We scored a good second goal but we were a little naive with the ones we conceded."
ROVERS: Usai, Henley (sub Daly 71 mins), Wylie, O'Connell (c), Beesley, Fernandez, R. Hanley (sub Haley 89 mins), O'Sullivan, Evans, Maclaren (sub Osawe 75 mins), Morris.
Subs not used: Urwin, Brown.