Youssouf Mulumbu’s 85th-minute winner gave under-pressure Pepe Mel his first victory as Albion boss and completed a remarkable turnaround.
Mulumbu netted a dramatic goal to hand the relegation-threatened Baggies their first win since New Year’s Day.
And it sealed an amazing change of fortunes after an abject first-half display that saw Mel’s men trail at half-time to Ronald Lamah’s second-minute strike.
The vultures appeared to the circling for Mel at the break after a dismal performance for his side, who could have trailed by more than a single goal.
But they improved dramatically in the second half and levelled through Stephane Sessegnon on 52 minutes before Mulumbu struck to send 2,000 travelling fans into delirium.
The only negative for the Baggies was a 10th booking of the season for Jonas Olsson that will see the defender suspended for the matches against Hull and Cardiff.
Mel took a gamble in his team selection by changing Albion’s formation to include three centre-backs with two defensive wing-backs in Steven Reid and Liam Ridgewell.
But the efforts to keep things tight were undermined in just the second minute when shocking Baggies defending handed the Swans the opening goal.
Wilfried Bony’s through-ball found Lamah, who burst behind Reid and then tricked his way past the recovering defender before producing a sliderule finish to beat Ben Foster.
The home side bossed the opening stages and the damage could have been worse for Albion on 18 minutes when Lamah found space on the Swansea left and cross for Bony, whose shot was kept out by a smart save from Foster.
Albion tested Michel Vorm in the Swansea goal when Victor Anichebe collected a smart pass from James Morrison and spun away from Ashley Williams, but his shot gave Vorm a comfortable save.
And the Swans might have doubled their lead before the half-hour mark when Wayne Routledge’s cross from the right just eluded Jonathan De Guzman at the far post.
They had a further chance when a cross from the left broke for De Guzman in the area but he stumbled under pressure from Craig Dawson and sent his shot wide.
And just before half-time more dismal Baggies defending allowed former Hawthorns trainee Ashley Williams a free header from a corner, but the Swansea defender sent his effort into the ground and over the bar.
Albion’s first half had been abysmal but they began the second half with much more urgency and might have levelled straight away when a corner was cleared as far as Youssouf Mulumbu, who saw his shot blocked by Jordi Amat.
And just seven minutes after the restart they were level when James Morrison fed Sessegnon, who took a touch before bending an excellent shot past Vorm.
Albion survived a huge penalty appeal when a shot on the turn from De Guzman struck the arm of Gareth McAuley, with replays suggesting a spot-kick would have been harsh on the visitors.
Anichebe then dragged a shot just off target after a neat Albion move on the edge of the Swansea area.
But the Baggies had a scare when De Guzman nipped in behind their defence and saw his shot saved by Foster’s feet.
With six minutes remaining Mulumbu struck to give the Baggies the lead after pouncing on a loose pass from Williams.
The Swansea defence opened up alarmingly for the Albion midfielder to stride through and net with a shot from the edge of the box that went in off the base of the post.