A Graham Dorrans penalty and a last-minute strike by Ishmael Miller secured an undeserved three points for the visitors.
The Baggies had a four-man midfield with Youssouf Mulumbu operating in front of the back four but the Swans still had the better of the opening exchanges with Darren Pratley causing problems with his darting runs through the middle.
There was little goalmouth action until midway through the half when a superb 40-yard pass from Ashley Williams set Nathan Dyer free down the right. His well-struck shot from the byline stuck the Baggies post with Scott Carson beaten.
Albion were irritated about the amount of possession enjoyed by the Swans and, after Jonas Olsson was given a stern ticking off by referee Rodger East, Dorrans went into the book on the half-hour after sliding down Angel Zaragoza Rangel.
Dyer should have broken the deadlock five minutes before the break when Mark Gower sent a ball across the goalmouth but he failed to get the touch needed and the chance was lost.
Four minutes after the break a strong run by Dorrans gave Luke Moore a sight of goal but Dorus de Vries was well-positioned to gather at his near post.
Apart from that it was the Swans who looked the better team and Moore was withdrawn in favour of Miller as the Baggies looked for more punch up front.
In the 78th minute Albion went ahead from the penalty spot in controversial fashion. Sub Giles Barnes raced into the area and seemed to dive following a challenge from Rangel but referee East pointed to the spot and Scotland international Dorrans scored off the inside of the post.
With five minutes remaining the Baggies were slowing the game down at every opportunity and, with the Swans pushing forward, a long ball from Dorrans left Miller racing through on his own and he rounded de Vries before finishing with a low shot to put the result beyond doubt.
It was a crushing blow for the Swans who deserved at least a point for their efforts but once again a lack of goals proved decisive.