Belgian winger Roland Lamah had put the home side in front early on as they enjoyed a dominant first half at the Liberty Stadium.
But Stephane Sessegnon equalised after the break and Youssouf Mulumbu was the hero when he fired home on 85 minutes to give Pepe Mel's side their first away victory since September .
Both sides were under pressure to pull themselves out of poor runs of results and pull away from the drop zone. Garry Monk's Swans were six matches without a win and the Baggies hadn't tasted victory since beating Newcastle 1-0 on New Year's Day.
Recharged and revved up after a 13-day break, the home side fired out of the blocks - taking less than two minutes to open the scoring.
Leon Britton turned in his own half and played the ball into Wilfried Bony. His perfectly-paced through-ball was latched onto by Osasuna loanee Lamah down the left.
When West Brom full-back Steven Reid came steaming into block what he thought would be a cross, Lamah cut back in-field and curled the ball into the far corner, past England international Ben Foster.
It was not until the 15th minute that the away support saw anything to excite them. James Morrison slid a pass into Victor Anichebe, who showed great strength to shrug off Ashley Williams before firing his shot straight at Michel Vorm.
Jonathan de Guzman came close when he met a Lamah cross, but Foster saved from point-blank range. Lamah was involved again when he had a shot blocked, then glanced a header wide.
Swansea continued to threaten as the half wore on. Bony sliced wide from a tight angle and De Guzman shinned the ball wide from a Lamah cross.
The visitors rarely kept hold of the ball long enough to make an impact on the game. The only times they threatened were when Swansea took a few risks trying to play their way out from the back.
Home skipper Williams nearly celebrated his 100th Premier League appearance for the club with a goal just before half time, but his downward header from a corner bounced up and over the bar.
West Brom emerged after the break with renewed intent, snapping at Swansea heels and hustling for the ball. The extra aggression paid off and in the 52nd minute midfielder Sessegnon levelled the match from 20 yards.
Bony lost the ball on half way, allowing the visitors to press and find James Morrison with his back to goal. He teed up Sessegnon, whose sweetly-struck drive beat Vorm.
Swansea almost took the lead again, but De Guzman's curling free kick was well held by the diving Foster. After his side had a hand-ball penalty shout turned down, Monk turned to a cult hero at the Liberty Stadium for some inspiration.
On 63 minutes, Spanish striker Michu - last season's top scorer - came off the bench to a standing ovation, making his first appearance since December 15 due to injury. He was not the hero this time, making little impact up front.
Both sides exchanged chances before The Baggies took the lead in the 85th-minute through Congolese midfielder Mulumbu. He picked up the ball 30 yards out and was able to drive towards the Swansea goal without anyone closing him down. His side-footed strike from just outside the box went under Jordi Amat's legs and in off the post.
Not since September 28 had West Brom won on the road, when they upset Manchester United 2-1 at Old Trafford.