Both of these sides entered the clash in good form and off the back of fine wins last week but, having won at Arsenal, it was Swansea who were the favourites at kick off. However, Chris Hughton's men had other ideas and they stormed out of the blocks and took a deserved lead on 16 minutes.
Former Rangers full back Steven Whittaker bombed forward and found himself in all sorts of space as he sent Gerhard Tremmel the wrong way with a left-footed drive from the edge of the box.
Swansea couldn't get themselves going in a tough first half for the hosts and it came as no great surprise when Norwich doubled the lead through Sebastien Bassong's powerful far post header five minutes before half time. It looked like game over three minutes later when Grant Holt was left unmarked to add another for the Canaries, who led comfortably at the break.
Back came Swansea as they second half began, though, and when in-form striker Michu cut the deficit to two just six minutes after the break with a simple tap-in it seemed to spark them into life. Jonathan De Guzman capitalised on a poor clearing header to half volley in their second on the hour mark and suddenly it was game on.
Norwich were on the rack but their one real chance of the second half was all they needed to restore the two-goal cushion. Robert Snodgrass, excellent throughout, curled a superb free-kick inside the near post to leave Tremmel motionless and send the travelling fans wild.
Michu headed home a third for Swansea in the dying embers of a superb match, but it was not enough as Norwich held on for their first win on the road this season.