The hooker was a late call-up into Anthony Foley’s starting 15 after Eusebio Guinazu had to withdraw and miss his Munster debut because of a tight hamstring.
But Casey made the most of his chance in the side with two tries in the second half to add to scores from Robin Copeland, Andrew Smith and Ronan O’Mahony.
Munster had been trailing 9-7 at half-time but a vastly improved second-half performance from Foley’s much-changed side saw them take the victory.
Foley made seven changes from the side who had edged past Cardiff Blues on Munster’s last trip to Wales in the Pro12 at the beginning of November.
JJ Hanrahan was deployed at full-back for the first time in this campaign and Denis Hurley, Ronan O’Mahony, Cathal Sheridan, Casey, Stephen Archer and Sean Dougall were all brought into the starting line-up.
The Dragons started the better of the two sides but failed to take advantage of a penalty inside the opening two minutes when Geraint Jones’ kick came back off the post.
Munster’s defence did well to hold up Dragons captain Rynard Landman on the line but when referee Matteo Liperini went back for a penalty, Jones made no mistake with the boot at the second attempt.
After the restart it was the visitors chance to put the hosts under some pressure and Ian Keatley pegged the Dragons back with a superb kick to the corner.
Munster had the Dragons in trouble from the line-out and after opting to go for the corner with a penalty number eight Robin Copeland, fresh from making his Ireland debut against Georgia on Sunday, was able to force his way over from close range.
Outside-half Keatley, who like Copeland was released back to the province from Ireland duty, added the extras with the boot.
Midway through the first half Jones clawed the Dragons back to within a point with a second penalty as the momentum continued to swing between the two sides.
The Dragons twice failed to make the most of good field position, first getting turned over when they had spare men out wide before conceding a penalty at the line-out just five metres from the Munster line.
They did eventually take the lead with a third penalty from Jones five minutes before the interval after Keatley was caught offside. When play resumed Keatley then put a second kick long over the dead ball line as he appeared to be feeling the pressure of the raucous Rodney Parade crowd.
But he had a chance to atone for his mistake two minutes after the interval when the Dragons were caught offside close to the halfway line and he slotted the penalty.
Munster were looking far more composed at the beginning of the second period and soon had the Dragons on the back foot.
Dragons scrum-half Ritchie Rees had a clearance kick charged down before the home side were penalised for offside.
Keatley kicked for the corner and the Dragons were unable to do anything about Munster’s driving line-out, with Casey crossing for his first try.
A conversion from Keatley extended Munster’s advantage to eight points and they began to take full control of the contest.
But Foley’s side were failing to make the most of their territorial dominance and with such a slender lead there was always a danger the Dragons could force their way back into the game.
There were a couple of anxious moments for Munster before centre Andrew Smith settled the nerves with their third try. This time they did make the most of an overlap out wide and Smith was able to break the line and race clear under the posts.
Keatley added the extras before Jones kicked his fourth penalty for the Dragons three minutes later to reduce the deficit to 12 points but it was little more than a brief moment of relief as Munster continued to dominate.
Casey wrapped up the bonus point with a second try from another driving line-out before Ronan O’Mahony grabbed a fifth in the dying moments.
Pens: R. Jones 4.
Tries: Copeland, Casey 2, Smith, R. O’Mahony. Cons: Keatley 5. Pens: Keatley.
G Jones, T Morgan, R Wardle, A Smith, H Amos, D Jones, R Rees (J Evans 60); B Harris, E Dee (T Thomas 79), D Way (L Fairbrother 65), C Hill, R Landman (capt), J Thomas, N Cudd (O Griffiths 64), Evans.
JJ Hanrahan; A Conway (Murphy 38), A Smith (P Howard 71), D Hurley, R O’Mahony; I Keatley, C Sheridan (D Williams 58); J Ryan (A Cotter 73), D Casey (N Scannell 73), S Archer (B Botha 73); D O’Callaghan (Dave O’Callaghan 73) (capt), B Holland; P Butler, S Dougall (C Stander 58), R Copeland.
M Liperini (FIR).