Chris Hughton's side raced to a three goal lead in the first half and almost let the Swan's get back on level terms but managed to hold on for its first away win of the season.
The Canaries raced into a three goal lead at halftime but allowed the home side to get back to 3-2 in the second half. Michael Laudrup's side pushed for the equalizer but Robert Snodgrass sealed the points with a late free kick. Michu grabbed a consolation goal late on but the Canaries hung to for the result.
Laudrup made two changes from the win at the Emirates, Danny Graham and Wayne Routledge both returned from injury, with Itay Schechter dropping to the bench and Leon Britton absent altogether. Chris Hughton brought Jonny Howson and Russell Martin back into his starting lineup, Alexander Tettey and Ryan Bennett made way.
The Swans began the game on the front foot but it was Norwich which had the first chance of the game. Robert Snodgrass's corner was flicked towards goal by Bradley Johnson but Gerhard Tremmel reacted well to keep it out.
Swansea continued to enjoy more possession but it was the Canaries, again, who came closer to scoring. Howson was unlucky to see his superbly struck shot from distance come back off the post, moments later Tremmel dropped Javier Garrido's cross but Snodgrass' failed to capitalize on the goalkeeper's mistake.
Hughton's side eventually made the breakthrough after 16 minutes when Whittaker made an excellent surging run forward before playing a one-two with Snodgrass and firing a powerful low shot into the near post.
Swansea's best chance of the half came just before the 30 minute mark, when Graham headed Jonathon De Guzman's cross on to the bar.
The Swans' continued to pile forward towards the end of the half as Norwich began to sit back, but the home side couldn't make the possession count and allowed the Canaries to double their lead in the 40th minute. Snodgrass' free kick was cleared only as far as Holt, who whipped in a cross for Bassong to head into the near corner past Tremmel, who should have done better.
Yet another set piece delivery from Snodgrass was met this time by Holt. The unmarked Norwich forward gave the keeper no chance with a bullet header across the goal into the corner.
It should have been four for Norwich minutes after the restart. Snodgrass lifted a ball over the Swansea defense for Johnson who almost scored a sublime goal with an exquisite volley but saw his effort hit the crossbar.
Swansea survived the scare and went straight back up the pitch to pull one back through Michu. The Spaniard swept home Davies' deflected cross into the far corner past Mark Bunn. Graham was slipped through on goal minutes later but could only direct his effort at the Norwich keeper.
Ten minutes later and the comeback was well and truly on, Dyer hooked in a cross which Routledge kept alive. The ball eventually dropped to De Guzman and, 10 yards out, the midfielder fired emphatically past the keeper.
At 3-2, all the pressure was coming from Swansea and the equalizer seemed inevitable, but Snodgrass once again punished the home side from a set peice. The former Leeds winger took to goal directly this time, swerving his effort over the wall and into the bottom corner.
Michu made it interesting late one when he turned in a De Guzman's cross in injury time, but it proved to be too little to late.