The hosts went about their task with real precision in the opening half, as an outstanding strike from Jonathan de Guzman gave them the lead in the 30th minute.
The Dutch midfielder grabbed himself a brace eight minutes later when he got on the end of Wilfried Bony's creative flick to slot past John Ruddy to make it 2-0 before the break.
Here, Sports Mole takes a closer look at the action between the two teams
On target: 7
On target: 4
Was the result fair?
Swansea thoroughly deserved the three points at the Liberty Stadium. They have found it tough going in recent weeks, but their performance this afternoon suggested that they have the ability to be playing in the top half of the league as opposed to the bottom. Garry Monk's side played with intensity from the start, pressing Norwich and leaving them with no time to think regardless of their position on the pitch. When the home side had the ball, they played with an enterprise you'd usually associate with Barcelona and a ruthlessness of a Manchester United side under Sir Alex Ferguson.
Swansea have often been known for their ability to move the ball, but have been undone this season by being unable to kill games off when required. This afternoon they showed all the hallmarks of a side capable of seeing through a victory. The hosts appeared well-drilled in every section of the pitch, with Bony leading the line and causing a whole number of problems for the visitors. Swansea didn't give Norwich a second to breathe and that resulted in the home side having the run of the game from start to finish.
Credit to Swansea for making the game look so easy, but Norwich failed to turn up this afternoon and this is something Chris Hughton will be particularly concerned by. For a team apparently looking to stave off relegation, the visitors were extremely poor. Defensively they were unable to communicate and looked overrun each and every time Swansea came forward. Bony won almost every challenge he had to make against Norwich's centre-backs, who were all at sea when the Ivory Coast striker had the ball at his feet. Hughton will also be asking for more from his strikers, who were devoid of any creativity and even failed to provide defensive support by not coming back to squeeze Swansea's midfield.
Sports Mole's man of the match
Despite De Guzman grabbing two goals for his side, Bony was by far the best player on the pitch. He showed great inventiveness by dropping deep to collect the ball and backed it up with pure strength to hold off his markers. He played a wonderful flick-on to find De Guzman for his second, while also providing a target for the initial cross that led to Swansea's opener. Elsewhere, the striker consistently found space and gave his side an outlet when in possession of the ball.
It's hard to pick on a team's manager when his players have failed to perform, but Hughton has made some odd decisions in the past few weeks. Today he decided to start Ricky van Wolfswinkel, despite the striker only getting seven shots on target all season, while the clinical threat of Gary Hooper remained on the bench for the majority of the game.
Jonathan Moss had an impressive game at the Liberty. He had a few tough decisions to make, especially when Alex Tettey brought down Routledge in the closing stages, but he did well to communicate with both captains and ease any frustrations as they appeared to be boiling.
Swansea: Swansea have grabbed three vital points this afternoon and they now face a Premier League fixture away at Hull City. Today's win could be vital given that they also have to face Chelsea and Southampton before the season is out.
Norwich: This result may have seriously hampered Norwich's hopes of avoiding a relegation dogfight as their final four games see them face Liverpool, Manchester United, Chelsea and Arsenal. Before that, however, they face pivotal clashes against West Bromwich Albion and Fulham as they look to grab more points to keep them away from the drop zone.