The Swans came from behind to claim their second away win of the Premier League campaign.
Marc-Antoine Fortune had put West Brom ahead but just a minute later Gylfi Sigurdsson levelled with a neat finish from Neil Taylor's cross.
Sigurdsson then set up Danny Graham for the winner as the Swans - who were pegged back by Chelsea the previous week in the dying seconds - held on through three minutes of added on time.
The Swans kept the same XI that drew at Chelsea last time.
Ashley Williams, released by the Baggies as a 16-year-old, started again against his former club while Luke Moore - another member of the Swans squad to have played for West Brom, was among the substitutes.
For the hosts, new signings Liam Ridgewell and Keith Andrews were on the bench.
West Brom: Ben Foster, Jonas Olsson (capt), James Morrison, Steven Reid, Jerome Thomas (Somen Tchoyi, 72), Graham Dorrans (Simon Cox, 72), Nicky Shorey, Youssouf Mulumbu (Paul Scharner, 64), Gareth McAuley, Peter Odemwingie, Marc-Antoine Fortune.
Subs: Marton Fulop, Liam Ridgewell, Keith Andrews, Billy Jones.
Swans: Michel Vorm, Angel Rangel, Steven Caulker, Ashley Williams (capt), Neil Taylor, Leon Britton, Joe Allen, Gylfi Sigurdsson, Nathan Dyer (Wayne Routledge, 87), Scott Sinclair, Danny Graham.
Subs: Gerhard Tremmel, Garry Monk, Kemy Agustien, Josh McEachran, Luke Moore, Leroy Lita.
It was the visitors who continued to enjoy the majority of possession, and when Sigurdsson was tripped by Jonas Olsson near the edge of the box. the Icelandic international curled his free-kick just wide.
Then, at the other end, the Baggies had a big shout for a penalty as Gareth McAuley met James Morrison's corner but Ashley Williams appeared to make contact with his hand. Referee Jonathan Moss, though, waved away the appeals.
James Morrison then saw this effort easily saved by Michel Vorm while Danny Graham reacted quickest to a loose ball before firing just over from an acute angle.
West Brom then went within inches of opening the scoring. Marc-Antoine fortune's header from a corner was goalbound before Sigurdsson headed off the line, and then Neil Taylor came to the rescue by throwing himself in front of the shot.
Vorm then showed fine reflexes by keeping Olsson's effort out before Scott Sinclair felt he should have been awarded a penalty after he went down in the box.
Half-time: West Brom 0 Swans 0
Neither side made a change at the break, and it was the visitors who had the first opportunity. Nathan Dyer intercepted Youssouf Mulumbu's pass and raced on towards goal before firing just over.
But just moments later, the hosts took the lead. After a corner was flicked on, Fortune controlled before firing low past Vorm in the 54th minute.
The home celebrations didn't last long though. A minute later, Sinclair fed Taylor down the left and his low cross found Sigurdsson, who side-footed past Ben Foster.
Things got better for Brendan's side. After Sigurdsson raced on, the midfielder delivered a superb curling cross into the box and Graham poked home into the bottom corner after 59 minutes.
The Swans continued to push on and Dyer went within inches of a third as Graham fooled Olsson with a dummy before squaring to the winger, but his shot was blocked and it bounced wide.
But the hosts then had a glorious chance to level as an attempted shot whizzed along the face of goal and fell to the unmarked Peter Odemwingie, but he fired well over from eight yards.
It became end-to-end stuff, with Dyer then just failing to make contact with Angel Rangel's pinpoint through ball before Sigurdsson saw his volley blocked by Olsson as the game entered the last five minutes.
As the game entered three minutes of stoppage time, hearts were beating faster and faster in the away end.
And when Fortune had a big chance to score, only the boot of Vorm denied the striker as it squirmed just wide.
But that was enough for the Swans, as they grabbed a second away win that could prove crucial come May.