Dorrans crashed home from the 12-yard mark after Angel Rangel was alleged to have fouled Giles Barnes 11 minutes from the end and Ishmael Miller finished things off late on.
Both sides spent most of Tuesday night's game exchanging neat passing moves, all of which were without any genuine outcome, so it was fitting that the deadlock was broken by a set-piece.
That paved the way for Dorrans to create a clincher for Miller as the Baggies moved eight points clear of Nottingham Forest in third and just a point behind Newcastle, who admittedly have played two games less.
After a disappointing draw at home to Sheffield Wednesday at the weekend, Swansea boss Paulo Sousa made four changes to his starting line-up and it was one of the new faces, captain Garry Monk, who spurned the first chance when he glanced a header wide from a free-kick.
Nathan Dyer, another to return to the fray, then curled an effort just wide from the edge of the area as Swansea pressed.
Mark Gower rattled a drive away from goal for the Swans as they continued to probe, but despite both sides impressing in possession, the fact that they were both playing with five midfielders made genuine chances hard to come by.
That was a point further illustrated just before the break when Rangel barely worked Scott Carson, his strike just the fifth of the opening half.
Baggies striker Luke Moore registered his side's first shot minutes into the second half and forced a smart, low save from Dorus de Dries, but the lack of opportunities being created by either side was almost painful to witness.
Swansea's Alan Tate attempted to liven things up with a speculative strike from 25 yards that did little to worry Carson, while the former England stopper was also on hand to field Dyer's hit.
West Brom began to gradually wake up, though, and Gabriel Tamas flicked a header wide as a statement of their intent.
It served as notice for their goal, too, with Dorrans slamming home from the spot.
The home side questioned referee Roger East's decision to award the kick after Rangel and Barnes came together in the box, but it mattered little to Dorrans as he tucked away his 15th goal of the season.
Rather than reignite Swansea's interest in the game, the goal initially served to fire West Brom, for whom Barnes went close from outside the box, before Miller coolly finished with a minute to go after being played in by the impressive Dorrans.