West Brom led through a Marc-Antoine Fortune goal eight minutes into the second, but allowed Swansea to rebound with two quick goals from Gylfi Sigurdsson and Danny Graham.
Roy Hodgson's men went into the match hovering a few points above the relegation zone and with just two wins under their belt all season at The Hawthorns.
Swansea, a point ahead of their opponents in the Premier League, however, had the worst away record of all 20 teams with just one win to their credit, coming over Aston Villa.
The Welsh side had the best of the early play, dominating possession and they went close to opening the score through Sigurdsson after four minutes and again after 15 minutes.
West Brom counter-attacked and Gareth McAuley headed narrowly wide from a Nicky Shorey cross and James Morrison tested Swansea goalkeeper Michael Vorm with a long-range effort after 22 minutes.
The home side in turn was having the best of the pressure as the first half wound down and both Peter Odemwingie and Jerome Thomas went close.
And it was The Baggies that went ahead nine minutes into the second half when Cameroon striker Fortune fired home from inside the six-yard box into the bottom left corner.
The Swansea reaction was immediate and devastating with two goals in the next four minutes.
Firstly, the industrious Sigurdsson slotted home from close range for the equaliser and then Graham made it 2-1, also from close in.
A stunned West Brom poured forward in search of an equaliser and Morrison pinged one effort over the bar before substitute Paul Scharner had a snap shot well saved by Worm.
But Swansea held on to take all three points and in so doing easing themselves away from the relegation zone.
Source: AFP
Source: AFP