The visitors kicked off attacking the River End goal and Jason Scotland had an off-target shot inside the first minute.
In the third minute Ashley Williams was lucky to escape a booking for a late challenge on Arturo Lupoli and Ferrie Bodde was equally lucky not to see a yellow card for his challenge on the same player.
After 12 minutes Mark Fotheringham produced the first shot on target, but it went straight at the Swansea keeper.
From the fifth Norwich free-kick in the visitors' half of the field, Leroy Lita headed over.
In the 27th minute Lita's shot was saved but a minute later Fotheringham touched Matty Pattison's throughball to Lupoli on the edge of the Swansea penalty area and his first-time right-foot shot beat the keeper low to his right.
David Marshall rescued Norwich when he tipped over a fierce shot from Jordi Gomez. When the resulting corner came in Swansea had a goal disallowed because Scotland was flagged offside two yards out.
However, Scotland scored the equaliser two minutes from half-time when Gary Doherty failed to cut out a right-wing cross from Angel Zaragoza Rangel and the number nine stretched out his right foot to send the ball past Marshall from four yards.
There was still time for Pattison to see his shot come back off a post before half-time.
Norwich conceded two goals inside the first three minutes of the second half.
An in-swinging right-wing corner from Gomez was headed on at the near post by Darren Pratley and David Bell could only steer the ball over Marshall with his head.
Fotheringham's wayward pass then set up the Swans for their third goal. Gomez back-heeled the ball to Bodde who ran on down the inside right channel and placed his right-foot shot between Marshall's legs from eight yards.
On the hour Norwich pulled a goal back when Fotheringham's attempted throughball was deflected into the net off Rangel.
It was the last touch for the Norwich skipper who then made way for Wes Hoolahan. The substitute had a chance to make an immediate impact, but his shot went straight at the keeper.
Bell's goalbound shot was blocked by Rangel and, although the home side enjoyed most of the possession, no clear-cut scoring chance was created and the Swans ran out worthy winners.