The Hammers squandered a three-goal advantage when they lost at Upton Park last month to Albion.
But they were well on the road to banking all three points when Mullins tragically put the ball into the back of his own net with 10 minutes left on the clock.
Striker Scott Dobie chipped a pinpoint ball into the heart of the Hammers defence where Mullens tried to clear the danger.
But the midfielder, instead of heading the ball over his own cross bar, looped it perfectly over keeper David James and into the back of the net.
West Ham opened the scoring in the 69th minute after a determined run by Marlon Harewood.
He was brought down on the edge of the area but not before he slipped a fine pass to Brian Deane who fired in a superb swerving shot.
Albion's diving keeper Russell Hoult got a hand to the ball but he could not stop it from flying into the far corner of his net.
In the first half Albion came close when Jason Koumas beat 2 players in a cracking run before forcing James into a save with his shot 15 yards.
Then, in the 21st minute, the Hammers threatened the Albion goal for the first time when Deane's header to a cross from Wayne Quinn was taken by Hoult.
It was then the turn of Albion with James O'Connor firing in a 20 yarder that was taken by James before play switched and Deane charged his way passed defender Sean Gregan only to send his snap shot just wide of the far post.