The hosts were quick off the mark when skipper Anthony Foley grabbed the first try of the game after only three minutes.
The Dragons scrum was disrupted in midfield, Lawlor took the ball on, Gary Connolly did well in support and Foley crashed over to the left of the posts.
Munster flyhalf Mossie Lawlor added the extras but it was not to be an easy ride for the home side as they were made to work all the way for the win by the Dragons.
Inspired by a concerted forward effort the visitors grabbed a try of their own through prop Adam Black just minutes later.
Rhys Thomas went close before Black burrowed over the line, but Craig Warlow's touchline conversion faded wide.
Lawlor extended the Munstemen's advantage with a penalty before the break and things then got worse for the Dragons when lock Peter Sidoli was sent to the sin-bin for preventing a quick ball.
However Munster, who included only Foley and back three Shaun Payne, Ian Dowling and Barry Murphy from the starting line-up for last Saturday's memorable Heineken Cup win against Sale, failed to capitlise with the extra man.
After the break Dragons skipper Jason Forster went close to grabbing their second score and then it was Munster's turn to cope with 14-men when Ian Dowling was yellow-carded for a deliberate knock on.
The Dragons edged nearer with a 64th-minute penalty form Warlow but in the end they could not lift themselves past the Celtic League pace-setters.