The Canaries had stunned their high-flying hosts with goals from Steven Whittaker, Sebastien Bassong and Grant Holt giving them a 3-0 half-time lead.
Swansea rallied through strikes from Michu and Jonathan De Guzman but Robert Snodgrass made sure of the win with an excellent free-kick before Michu netted again in injury time.
There were warning signs inside the opening seconds as Bradley Johnson met Snodgrass' corner to force a save from Gerhard Tremmel. The pressure built as Jonny Howson rattled the base of a post and the Canaries took a deserved lead in the 16th minute as Snodgrass laid the ball off to Whittaker, who cut in and fired low past Tremmel.
Swansea's first genuine chance came when Danny Graham headed off the bar while Michu almost lucked into an equaliser five minutes later, the Spaniard's drive took a huge deflection off Wayne Routledge to leave Mark Bunn stranded but the ball spun just wide of his right-hand post.
Swansea were stopped in their tracks as Bassong struck for the third time in four games, rising above Ben Davies to beat Tremmel at the German's near post. Just two minutes later it was 3-0 as Snodgrass' perfectly delivered free-kick picked out an unmarked Holt, who easily steered his header beyond Tremmel.
Michu gave Swansea a route back into the game as he slotted into the far corner from Davies' low ball across the box in the 51st minute. The Spaniard almost turned provider for Graham and Bunn had to make a sharp save to deny the striker.
Swansea got back within a goal after 59 minutes as Nathan Dyer outpaced Javier Garrido down the flank and when his cross was not cleared De Guzman volleyed in. Graham could have levelled eight minutes later but Garrido did superbly to intervene as the striker ran in on goal.
His replacement Itay Shechter had the ball in the net soon after as Bunn failed to hold the ball under pressure from Michu but referee Howard Webb ruled the Spaniard had fouled the keeper.
Norwich doused Swansea's fire with their fourth of the afternoon as Ashley Williams was adjudged to have hauled down Holt. Snodgrass stepped up to curl home a beautiful strike from the resulting free-kick and secure the points for Norwich, meaning Michu's second in injury time was not enough for Swansea.