The Irish outfit were slow into their stride and missed the opportunity to take the lead when Ian Keatley was wide with a fifteenth minute penalty.
Instead it was the Dragons who got their noses in front after Steffan Jones made no mistake with his first attempt at goal six minutes later.
Keatley then atoned for his earlier error by levelling the scores with a 23rd minute strike before Munster scored the first try of the evening.
Danny Barnes found his way over the whitewash and Keatley converted to put Munster 10-3 in front.
A well-executed line-out move set up Munster for their second try, Lifeimi Mafi crossing under the posts with Keatley again supplying the extras.
Jason Tovey reduced the arrears by three at the end of the first period and two further kicks in the second half put the Dragons within one score of pulling off a shock result.
A third yellow card of the evening - handed to replacement front row Damian Varley for slapping an opponent - made for an anxious end to the game for the home faithful.
But a second penalty from Keatley put Munster more than two scores in front and, sadly for the Dragons, they left Ireland with nothing to show for their efforts when a long-range penalty from Tovey fell short with the last kick of the game.
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