De Guzman struck twice in quick succession late in the first half and Wayne Routledge added the third after the break to give Garry Monk’s side their first victory in 10 games in all competitions and end a six-match winless run in the Premier League.
De Guzman had been controversially denied the chance for a potential winner at Arsenal on Tuesday when referee Lee Probert blew for full-time in the 2-2 draw just as the Dutch midfielder was about to shoot.
The Dutch international made up for that frustrating finale at the Liberty Stadium as the Swans leapfrogged Norwich and moved seven points clear of the relegation zone.
Norwich’s seventh successive away defeat in all competitions left Chris Hughton’s team six points above the bottom three.
Monk’s team monopolised possession in the early stages without making the most of their domination.
Spanish striker Michu, back from injury recently, showed a little rust when he reached for Jonjo Shelvey’s cross and stabbed a shot well wide.
But Norwich had only themselves to blame when the hosts broke the deadlock in the 30th minute.
Shelvey’s cross was poorly cleared to the edge of the area, where the unmarked de Guzman was waiting to lash a low strike past John Ruddy.
De Guzman wasn’t finished yet and he capped an impressive move to double Swansea’s lead eight minutes later.
Wilfried Bony was the instigator with a clever back-heeled pass that set de Guzman clear and the midfielder took full advantage to clip his shot over Ruddy from eight yards.
The Canaries desperately needed an early goal in the second half and they almost got it when Robert Snodgrass’s effort from outside the area was headed off the line by Ashley Williams.
Monk’s men responded with a sweeping attack which nearly culminated in a goal when de Guzman unselfishly played in Shelvey and the midfielder slipped his marker before shooting narrowly wide.
Swansea put the result beyond doubt in the 75th minute when Shelvey played the ball to Routledge on the left side of the area and the former Crystal Palace winger fired home.