Craig Beattie made a fairytale return to his former club by scoring the goal that sunk the league leaders at the Hawthorns.
The Scottish striker, who made the switch from the West Midlands back in August, struck five minutes into the second-half following a well worked move by the visiting side.
Following an even first-half the Swans burst out of the blocks to take the lead but had no soak up plenty of Albion pressure.
West Brom piled on the pressure but the visitors remained strong and did have chances to extend their lead, including a strike from Tom Butler which was pushed onto the post.
But the Swans held on to register the win and make it seven Championship matches unbeaten.
Swans boss Paulo Sousa made two changes to his side for the clash at league leaders West Brom.
Marcos Painter came in to replace Fede Bessone on the left-hand side of defence while Cedric van der Gun started in midfield in place of the injured Andrea Orlandi (thigh).
Joe Allen, meanwhile, was passed fit to take his place amongst the substitutes.
West Brom: Scott Carson (C), Gianni Zuiverloon, Joe Mattock (Simon Cox, 75'), Jonas Olsson, Gonzalo Jara, Marek Cech (Filipe Teixeira, 46'), Graham Dorrans, Youssouf Mulumbu, Jerome Thomas, Chris Woods, Luke Moore (Robert Koren, 46').
Not Used: Dean Kiely (GK), Abdoulaye Meite, Shelton Martis, Reuben Reid.
Swans: Dorus de Vries; Angel Rangel, Ashley Williams, Alan Tate(C), Marcos Painter (Jazz Richards, 46'); Leon Britton, Darren Pratley, Mark Gower (Joe Allen, 75'), Cedric van der Gun, Nathan Dyer (Tom Butler, 62'); Craig Beattie.
Not Used:David Cornell (GK), Jordi Lopez, Lee Trundle, Besian Idrizaj.
Referee: A D'Urso (Essex)
After a 15 minute delay this fixture finally got underway with both sets of supporters making themselves heard.
The home side were on the offensive instantly with Jerome Thomas finding space wide on the right and crossing into the heart of the area, only for Dorus de Vries to coolly collect in mid air.
But the Swans didn't make this trip to the West Midlands just to sit back, and with just 90 seconds played Angel Rangel broke forward before playing in Nahan Dyer; the tricky winger cut inside but his effort from distance sailed high and wide of the Baggies' goal.
Graham Dorrans fired a similar effort high into the stand containing both West Brom and Swans fans while Mark Gower played to the feet of Darren Pratley in a dangerous position inside the area only for it to trickle away from his feet and out of play.
Dorrans fired over again before that while on 12 minutes the one-cap Scottish international drilled the ball over as Marek Cech flicked a free-kick into his path.
The Swans countered the early pressure and started to flow the ball steadily out of defence.
Angel Rangel looked confident going forward and had the pace of Dyer to his advantage in front of him. And with 20 minutes on the clock it was a dangerous cross from Dyer which could have seen the visitors open the scoring against the run of play.
His wicked right-sided cross evaded Scott Carson and his defence, as well as Darren Pratley six-yards from goal, as the ball whizzed right across the area.
Six minutes later and the Baggies were awarded a second free-kick right outside the area - the pace of Thomas getting the better of skipper Alan Tate - but he picked himself up to send a poor set-piece over the top of goal.
But the best chance of the half fell to the Swans on the half-hour mark when ex-Baggie Beattie capitalised on a poor clearance to burst into the box and reach the byline before squaring across to Dyer whose first time effort was blocked by Carson.
Paulo's side were given a double scare minutes later as de Vries failed to collect Cech's right-sided free-kick, Tate cleared as far as Gonzalo Jara and his deep ball into the heart of the area was met by the outstretched boot of Chris Wood who, fortunately, guided the ball over from five-yards.
A fine counter attacking move down the right saw the Swans win their first corner of the night and from Mark Gower's delivery Ashley Williams popped up with a powerful header and Carson needed to use his sharp reflexes to stop the ball on the line.
Cedric van der Gun, meanwhile, cut past three players inside the box only to send the next effort over the top of goal from close-range five minutes before the break.
The Swans played out the remainder of the half by playing a lot of keep ball and with a minute remaining until the break won their second corner.
Gower delivered into the heart of the area and Williams directed a strong header at Carson but it was West Brom that had the final word with Luke Moore latching on to the end of a long ball forward but firing over the top.
Half-Time: West Brom 0 Swans 0
Paulo made one change at the start of the second period as Marcos Painter, who was booked towards the end of the first-half, was replaced by youngster Jazz Richards, while the home side made two changes of their own.
The hosts made a strong start to the half with Dorus standing strong to take two deliveries into the area, but it was a counter attacking move on 50 minutes which put the visitors in front.
A fine run by Cedric van der Gun down the left saw him reach the edge of the Baggies' area, he found Gower who in turn played in Craig Beattie - the former West Brom man looked cool and composed as he took a touch before drilling the ball past Carson to make it 1-0.
West Brom responded as they went looking for an immediate leveller but it was the Swans that could easily have added to their tally just two minutes later - it was three against three with van der Gun the last on the ball, he took a touch too many inside the area before losing the ball.
Paulo's side were having to deal with a bit of West Brom pressure and just before the hour Dorus de Vries was called into action to make a save from Gianni Zuiverloon's strike from outside the area.
Nathan Dyer had been a thorn in the side of West Brom's defence all evening but his game was brought to an end on 62 minutes as Tom Butler was introduced to replace him out wide.
The pressure was back on the visitors but they stayed strong and countered it well - the Baggies were awarded a free-kick in dangerous territory only to waste it as Ashley Williams headed clear Graham Dorran's delivery, while Chris Wood shot low to force a save from de Vries on 69 minutes.
The Swans made their third and final chance with 15 minutes to go as Joe Allen made his return from a hamstring injury, replacing Mark Gower, to make only his third appearance of the season.
Simon Cox was also introduced by the hosts and after Darren Pratley lost possession in midfield the striker was through, played a low shot past de Vries but couldn't hit the target.
Albion continued to pressurise with a string of constant attacks but an ambitious Tom Butler strike from distance showed that the Swans were not just going to sit back and welcome the pressure.
That move was orchestrated by Joe Allen and the young Welsh international had a chance of his own two minutes later but was way off target.
But an excellent move started by Butler saw van der Gun and Beattie play the Irishman back in, he charged down on goal and with his strike forced Carson into saving and pushing his effort onto the post.
The Swans had to soak up the pressure in the closing minutes but held on to record an excellent victory and extend their unbeaten run to seven matches,