It all went wrong for United after they lost skipper Andy Crosby through injury at half-time.
It was Crosby who fired his side in front in the 16th minute with his third goal in successive home games.
It came from the penalty spot when he confidently beat goalkeeper Dean Kiely after full-back Carl Hoefkins had been penalised for his challenge on Andy Butler as a corner came into a crowded box.
Scunthorpe could have made it 2-0 when a curling 25-yard shot from striker Martin Paterson finished just a whisker wide.
Albion's first real effort came late in the half when striker Ishmael Miller blasted wide from the edge of the box when he should have done better.
Miller almost got an equaliser on the stroke of half-time when he rattled the outside of a post with an angled shot after Scunthorpe had been caught in possession.
While United were still trying to reorganise following the half-time loss of Crosby, it was Albion that came storming out to seize command at the start of the second half.
They equalised in the 49th minute when centre-half Leon Barnett crashed home a shot from close range after United had failed to clear a corner.
Albion went in front on the hour when a shot from Miller was blocked by keeper Joe Murphy, but only as far as winger Chris Brunt who steered his shot into an empty net.
Scunthorpe tried to battle back and twice came within a whisker of an equaliser.
First, winger Kevan Hurst blazed wide from the edge of the box and then midfield man Matt Sparrow lifted his shot over from a similar position.
It was Albion that made the game safe in the 78th minute when Portuguese star Felipe Teixeira won possession outside the box and forced his way inside to crash a good shot past Murphy.
But Scunthorpe were not finished and substitute Jonathan Forte headed against a post from close range before leading marksman Paterson reduced the arrears in stoppage time with his fifth goal of the season.