The successful spell for Norwich also sees them go 10 matches unbeaten in all competitions and it is a remarkable statistic for a team who had been tipped for relegation at the beginning of the campaign.
It was a dream first half for Chris Hughton's visitors as they took a three-goal lead through Steven Whittaker's solo effort and headers from Sebastien Bassong, his third goal in four games, and Grant Holt.
Michael Laudrup's Swansea, though, emerged a team with intent following half-time and scored twice before the hour mark through Michu and Jonathan de Guzman to set up a dramatic final half-an-hour in Wales.
But Norwich could breathe a little easier when the excellent Robert Snodgrass curled in a fine free-kick to mean Michu's injury-time header gave Swansea only a small reason to cheer before the final whistle.
The Canaries took a deserved lead in the 16th minute, when Snodgrass laid the ball off to Whittaker and the ex-Rangers man was allowed to run before cutting in and firing low past Gerhard Tremmel for his first Norwich goal.
Swansea had been building momentum but they were stopped in their tracks as Bassong met Holt's 40th minute cross to make it 2-0.
And just two minutes later, it was 3-0, with Snodgrass' perfectly delivered free-kick picking out an unmarked Holt, who easily steered his header beyond Tremmel to cap a superb first half for the Canaries.
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.
And Norwich's nerves only increased as Swansea got back within a goal on 59 minutes, when Nathan Dyer outpaced Javier Garrido down the flank and, when his cross was not cleared, De Guzman volleyed in.
Substitute Itay Shechter had the ball in the net soon after as Mark Bunn failed to hold the ball under pressure from Michu but referee Howard Webb ruled the Spaniard had fouled the goalkeeper.
Swansea's hopes were then over when Snodgrass stepped up to curl home a beautiful strike from a 77th minute free-kick, meaning Michu's second in injury time was not enough.