Routledge, who had a loan spell with the Bluebirds five years ago, did the damage against his former club on Saturday evening as Monk began his spell as Swans manager with a much-needed three points.
The result at the Liberty Stadium moved the hosts up to 10th in the Premier League table, four points clear of the drop zone where their beaten opponents remain. Cardiff are 19th, three points from safety.
After a dreary first half, Monk introduced Pablo Hernandez to the action for the second period and within two minutes, the Spaniard had sent Routledge racing away down the left to score past David Marshall.
Craig Bellamy hit the crossbar for the visitors soon afterwards, but they were undone once more as Routledge crossed from the left in the 79th minute for Nathan Dyer to stoop low and direct a header into the corner.
A third goal followed five minutes from time for record signing Wilfried Bony, heading home from close range after Pablo's free-kick.
Monk included Chico Flores in his first Swansea line-up, while Michel Vorm returned in goal. Flores was recently involved in a training-ground row with Monk - installed as head coach after the dismissal earlier in the week of Michael Laudrup - but partnered captain Ashley Williams in defence, with loan signing Marvin Emnes handed his first start since joining from Middlesbrough.
Cardiff boss Ole Gunnar Solskjaer handed Wilfried Zaha a first start for the Bluebirds after the winger's impressive debut off the bench against Norwich, and Kim Bo-kyung was also included as Jordon Mutch and Craig Noone dropped to the bench.
After just 44 seconds, Marshall was forced into action when Emnes beat Ben Turner and broke forward to free Routledge, whose tame shot was easy for the Cardiff keeper.
Two minutes later, Marshall made a superb challenge on Bony as Routledge's reverse pass picked out the bustling striker, the keeper pushing the ball clear as the Ivorian tried to round him.
Kenwyne Jones should have given Cardiff the lead in the 14th minute. Bellamy dug out an excellent cross from the byline but the Trinidadian headed wastefully wide.
A better chance arrived soon after. Peter Whittingham controlled a poor clearance and burst towards Michel Vorm's goal, only to slip at the crucial moment.
Another surging Bellamy run picked Swansea apart, but again Jones could not profit from the veteran's good work, prodding wide.
Vorm saved a dipping Whittingham volley as, while being penned back, the visitors had the better sights of goal.
The second half was under way for barely a minute before the opener arrived.
Pablo, on for Emnes at the interval, threaded a pass beyond a dozing Fabio for Routledge to race into the box and beat Marshall for his first league goal of the season.
Cardiff's response was instant, with Bellamy clattering the crossbar after coming in from the left and unleashing a powerful drive, while Kim had a penalty appeal turned away as he went to ground under Davies' challenge.
Dyer's dribble ended with the winger bending an effort narrowly wide as the contest opened up.
Cardiff should have been level with 12 minutes left. Bellamy's free-kick struck Turner and fell to Kim, but the Korean hooked his volley over the bar.
The miss proved costly as Swansea quickly killed the game off. Routledge and Pablo led a breakout, with the former's left-wing cross dropping perfectly for Dyer to head low into the corner of the net.
And Bony capped off the win by powering a header into the roof of the net from Pablo's free-kick to claim his 17th goal of the season, allowing Monk to savour the delight of the home support at the final whistle.