The men from south Wales had gone nine matches without a win since a 3-0 triumph over local rivals Cardiff City in Monk’s first game in charge, but a Jonathan de Guzman brace broke that bad run and lifted Swansea seven points clear of the relegation zone.
The hosts were dominant throughout and took the lead on the half-hour mark, when De Guzman found the net with a powerful drive from 18 yards.
And the Netherlands international doubled his tally before the break, putting Swansea in firm control, before Wayne Routledge rounded off a well-deserved victory.
As well as easing any lingering fears of relegation for Monk’s men, the result leaves Norwich six points clear of the drop zone, having played three games more than 19th-placed Sunderland.
Leon Britton made his 400th league appearance and the midfielder was in the thick of the action as Swansea dictated the pace in the early stages.
Their first chance fell to Michu, who recently returned from a three-month injury layoff and showed signs of rustiness as he scuffed his effort wide of the right-hand post from the edge of the penalty area.
The Norwich defence was having to withstand a considerable amount of pressure, but Chico Flores almost put the ball in his own net at the other end in the 19th minute as his wild swipe at a cross from the right flew over the crossbar.
Michu slid in to guide the ball just wide of the target from close range five minutes later, before John Ruddy got down well to deny Routledge.
The Norwich goalkeeper palmed the ball into the path of an advancing De Guzman, but Wes Hoolahan made a superb challenge to deny the Dutchman a glorious opportunity.
Swansea’s domination paid off when De Guzman sent a sweet left-footed strike into the bottom right-hand corner of the net from the edge of the box.
De Guzman then had a delightful Wilfried Bony flick to thank for teeing up his second eight minutes later – the Ivorian frontman leaving his team-mate with the simple task of prodding the ball beyond Ruddy for his seventh goal of the season in all competitions.
Norwich burst out of the blocks after the break and almost caught Swansea by surprise, with Ashley Williams needing to clear off the line from a Robert Snodgrass strike.
The visitors continued to struggle to get a foothold in the match, although Michel Vorm had to be in the right place at the right time to deny Johan Elmander from point-blank range.
Routledge put the result beyond doubt in the 75th minute, latching onto a Jonjo Shelvey pass to slot the ball beyond Ruddy.
Bony then repeated the flick that led to De Guzman’s second goal 10 minutes from time, causing further chaos in the Norwich box with the ball eventually ricocheting off Sebastien Bassong and hitting the post.
