The hosts went ahead at the Liberty Stadium within the opening two minutes – Roland Lamah cutting inside to bury the ball in the bottom right-hand corner.
Swansea dominated possession throughout the first half, but saw their lead cancelled out seven minutes into the second when Stephane Sessegnon found the net from 20 yards.
And the comeback was complete in the 85th minute when Youssouf Mulumbu guided the ball into the bottom right-hand corner to earn the visitors their first win in 10 in all competitions - nine of which were under Mel.
Victory lifts West Brom three points clear of the relegation zone, while Swansea still have just a four-point cushion.
Swansea were boosted by the return of Michu to the bench after three months out with an ankle injury and the Welsh side gave their visitors no time to settle, capitalising on some poor West Brom defending to surge into a second-minute lead.
Wilfried Bony threaded the ball to Lamah on the left wing and the Belgian cut inside, left Steven Reid for dead at the edge of the penalty area, before coolly slotting the ball beyond Ben Foster for his second league goal for the club.
West Brom posed little threat until Victor Anichebe got the better of Ashley Williams – making his 100th Premier League appearance – on 15 minutes, but he fired his effort straight at home goalkeeper Michel Vorm.
Moments later, Foster found himself in the right place at the right time to somewhat fortuitously deny Bony from a Lamah cross.
Lamah looked bright throughout the opening period and headed just wide of the right-hand post from a Jonathan de Guzman corner midway through the half.
Swansea may not have reproduced the scintillating performance that saw them hold a 3-1 half-time lead in this fixture last season, but West Brom were showing few signs of being able to get back into the game as the hosts continued to dominate possession and look the more threatening.
But the visitors came out with renewed energy after the break and pulled themselves level with a low 20-yard drive from Sessegnon in the 52nd minute.
De Guzman then forced Foster into a diving save from a free-kick as Swansea looked to re-establish their advantage, before having an effort blocked by the arm of Gareth McAuley in the West Brom area, but referee Martin Atkinson waved play on.
The match looked to be heading for a draw, until Mulumbu saw the Swansea defence open up in front of him, allowing him the time and space to find the bottom right-hand corner of the net five minutes from time.
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