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Scotland international midfielder Dorrans again ran the show for third-placed Albion.
He fired them ahead from the penalty spot in the 19th minute and set up the crucial third goal for Jerome Thomas as Albion picked up their first Championship win in four games to close the gap on second placed Nottingham Forest to two points.
Roman Bednar had put Albion firmly in control with their second on the half hour before United threatened to mount a comeback when Darius Henderson's 49th minute penalty handed them a lifeline.
But United, who had not conceded a goal in winning their previous three Championship matches, saw their recovery hopes snuffed out just 53 seconds later by Thomas.
Things could have been a lot different for United had Richard Cresswell not missed a golden chance in the 10th minute after he was denied by a smart save from Scott Carson when he only had the England goalkeeper to beat.
It was a pivotal moment as Albion took charge nine minutes later when referee Mike Russell pointed to the spot after ruling somewhat harshly that Nick Montgomery had handled the ball under pressure from Dorrans.
The decision sparked a furious reaction from United but when the spot kick was eventually taken, Dorrans, who recently signed a new three-and-a-half year contract, sent Mark Bunn the wrong way for his 11th goal of the season.
Bednar, who had scored twice against United at Bramall Lane in August, should then have doubled Albion's advantage only to slice wide from six yards.
But he soon made amends when United were cut apart by a quick Albion break.
Bednar collected a cross-field from Gonzalo Jara on the halfway line and then picked out Thomas.
He continued his run forward to collect a return pass from Thomas before sliding a low shot just inside Bunn's near post.
Bednar was then almost gifted his second goal four minutes before half-time when he intercepted Stephen Quinn's under-hit back pass only to slice his shot into the side-netting.
Albion were cruising at this stage but did their best to gift United a route back into the game at the start of the second half.
A misplaced header by Marek Cech led to a spell of United pressure that ended with Quinn before hauled down in the area by Youssouf Mulumbu.
Henderson drilled his 11th goal of the campaign past Carson but United's fans had hardly got their celebrations into full swing before Dorrans long ball up field released
Thomas and his mazy run ended with his curling drive flying beyond Bunn.