Paul Lambert's Canaries got off to a perfect beginning when they hit the front inside the opening minute when Anthony Pilkington fired home after Steve Morison's header from an Elliott Bennett cross fell at his feet.
Less than ten minutes later and the Canaries were two to the good when Russell Martin escaped his marker in the area to head home David Fox's free-kick.
Swansea were rocked but were able to reply in the 12th minute when Danny Graham, who scored last time out, finished off Scott Sinclair's excellent piece of work down the left to give Brendan Rodgers' men some sort of hope early on.
But Pilkington quashed any hope of a Swansea revival when he netted his second just after the hour-mark with a neat swivel and finish from Bradley Johnson's pass to send Norwich into the top half, replacing Swansea in ninth position on the table.
Yellow cards:NOR - nil SWA - Garry Monk 4'