Norwich led 3-0 at half-time and looked on course for a straightforward victory, but goals from Michu and Jonathan De Guzman brought hosts Swansea back into the game after the interval.
Robert Snodgrass' excellent free-kick put the visitors 4-2 up with 13 minutes remaining, before Michu found the net again in injury time to become the Premier League's outright top scorer with 12 goals.
Victory extended Norwich's run of matches without defeat in the Premier League to nine games and took Chris Hughton's side up to 11th in the table, with Swansea sliding one place to eighth.
Swansea enjoyed a fine win at Arsenal last weekend but they were quickly forced onto the back foot by Norwich at the Liberty Stadium.
Jonny Howson saw a 20-yard strike come back off the base of the post in the 14th minute but within two minutes, the visitors were ahead.
Steven Whittaker exchanged passes with Snodgrass before stepping inside Swans captain Ashley Williams and beating goalkeeper Gerhard Tremmel with a low shot.
Snodgrass then forced Tremmel into action, before a brief Swansea rally suggested an equaliser might be imminent.
In the 28th minute, the recalled Danny Graham saw his header from a de Guzman cross hit the crossbar, while Michu had a deflected effort roll narrowly wide of the far post.
Swansea's flourish proved short-lived, however, as a quick-fire pair of goals from Norwich took the game away from the home side.
Sebastien Bassong made it 2-0 on 40 minutes, converting a cross from Grant Holt to score his third goal in four matches.
Tremmel was partly at fault for the goal but the entire Swansea defence went walkabout four minutes later, as Holt converted Snodgrass' free-kick with a powerful header for his first goal in nine games.
Bradley Johnson feathered a chip against the Swansea bar early in the second half, but any thoughts of a comfortable afternoon were eroded in the 51st minute when Michu pulled a goal back for the hosts.
After a typically patient build-up, Ben Davies put a low cross into the box from the left that the Spanish forward finished with one touch.
De Guzman's 59th-minute volley took Swansea to the brink of a rousing comeback, only for Snodgrass to restore Norwich's two-goal cushion with a fine free-kick from 25 yards.
Michu clawed another goal back for Swansea in injury time, but by then it was too late.