Norwich succeeded where the likes of Chelsea, Tottenham Hotspur and Arsenal failed by becoming only the second team to win at the Liberty this season.
The Swans had led through Danny Graham's first-haf goal but the visitors stunned the Liberty as Grant Holt levelled three minutes ito the second half before Anthony Pilkington's deflected strike off Neil Taylor put Norwich ahead.
It got worse for the Swans as Holt scored his second and the Canaries's third before Graham pulled one back from the penalty spot after Holt had tugged Ashley Williams's shirt in the box.
It set up a tense finish, with Steven Caulker twice being denied in added on time, but the Swans couldn't find an equaliser.
The Swans made one enforced change to their starting XI.
Joe Allen missed out as he was suffering with a hamstring injury, which meant Josh McEachran made his first Premier League start since arriving on loan from Chelsea.
Apart from that change, Brendan called Mark Gower onto the bench while Kemy Agustien was unavailable through injury in what was a largely unchanged squad from the one that won 2-1 at West Brom last time out.
Norwich were without centre-backs Zak Whitbread and Daniel Ayala, so full-back Russell Martin and Elliott Ward paired up in the heart of the visitors' defence.
Swans: Michel Vorm, Angel Rangel, Steven Caulker, Ashley Williams (capt), Neil Taylor, Leon Britton, Josh McEachran (Mark Gower, 64), Gylfi Sigurdsson, Nathan Dyer (Leroy Lita, 74), Scott Sinclair (Wayne Routledge, 74), Danny Graham.
Subs: Gerhard Tremmel, Jazz Richards, Garry Monk, Luke Moore.
Norwich: John Ruddy, Russell Martin. Adam Drury (Leon Barnett, 57), Grant Holt (capt), Simeon Jackson (Steve Morison, 83), Andrew Surman, Anthony Pilkington, David Fox, Elliott Bennett (Wes Hoolahan, 77), Elliott Ward, Kyle Naughton.
Subs: Ned Steer, Bradley Johnson, Andrew Crofts, Aaron Wilbraham.
Referee: Martin Atkinson
The game started brightly with the hosts creating the first opportunity within five minutes as Neil Taylor charged down Kyle Naughton's clearance and keeper John Ruddy sliced the loose ball to Gylfi Sigurdsson just in side the box.
But as he waited for the ball to drop, the Norwich defence regathered and got back in time to block his effort.
Both teams seemed to cancel each other out for the next 15 minutes, with neither side managing to force a way through.
Skipper Grant Holt did test Michel Vorm with a low drive, but that was easily saved by the Dutch keeper.
Then, on 23 minutes, came some fantastic vision by the hosts. Sigurdsson found Nathan Dyer 25 yards out, and the winger slipped through a pass for Graham, who controlled before firing past John Ruddy and into the top right-hand corner.
Anthony Pilkington, who scored twice in the earlier fixture at Carrow Road, then saw his shot deflected wide before Elliott Ward fired over from the edge of the area.
The visitors came again, Andrew Surman meeting a cross eight yards out, but Vorm made a fantastic reflex save.
The Swans responded, and almost doubled thier lead as Dyer slipped through another fine pass to Scott Sinclair on the left. He crossed with his first touch to Graham, but the striker was denied with a well-timed block some ten yards out.
The game was heating up nicely before the break, but Canaries full-back Adam Drury got too caught up in things as he was booked for a foul on Dyer.
Half-time: Swans 1 Norwich 0
The game was level three minutes after the restart. Russell Martin's volley flew wide, but Ward's cutback found Holt, and the striker headed back across Vorm and into the net.
The visiting supporters were still celebrating when Graham latched onto another through ball and rounded Ruddy. But the keeper reacted quickly to snuff his effort out for a corner. And from Sigurdsson's corner, Steven Caulker rose highest but saw his header rebound off the post.
The Swans had gone close to regaining the lead, but just moments later the visitors were ahead.
Simeon Jackson teed up Pilkington, and his effort took a massive deflection off Taylor and into the net.
The Liberty was stunned by the visitors' fightback, but the Swans had a great chance to level as Dyer's cross met Sinclair at the far post, but he header over.
Then, just after the hour, Norwich surged forward again and when Elliott Bennett found Holt unmarked, he fired in from an acute angle for a third.
Brendan immediately made a change, Mark Gower replacing Josh McEachran.
A last roll of the dice came with 13 minutes left, with Wayne Routledge and Leroy Lita replacing Scott Sinclair and Nathan Dyer.
But it was the visitors who went within an inch of scoring the next goal as Naughton saw his thunderous effort tipped onto the crossbar by Vorm.
The Swans then saw a penalty appeal turned down as Routledge's cross appeared to be handled by Naughton, but referee Martin Atkinson turned down the claims.
While the hosts were started to throw everything at Norwich, it was the visitors who almost made it 4-1 as Holt squared to sub Wes Hoolahan but the midfielder's point-blank effort was tipped over by Vorm.
Sigurdsson almost pulled one back with a free-kick but Ruddy tipped it over. From the resulting corner, the Swans had a lifeline as Holt tugged back Ashley Williams and Atkinson awarded the penalty.
With Sinclair not on the pitch, Graham took over penalty duties and sent Ruddy the wrong way with four minutes left.
It set up a tense finish, but Norwich were still looking for another goal.
They almost got it when sub Steve Morison was in on goal down the left, but Vorm denied him again with a spectacular save low down.
Five minutes off added on time was set to be played, with Graham just missing out on an equaliser - and his hat-trick - as he sidefooted Sigurdsson's low cross just wide.
Caulker was then denied bya fantastic Ruddy save from Sigurdsson's corner deep into added on time, before the defender then saw his shot deflect off Lita and bounce wide.