A first-half try by skipper Andy Marinos and three penalties and a conversion from the boot of Percy Montgomery did the trick. Munster's points came from a brace of penalties from winger Mossie Lawlor.
The victory sees the Dragons still four points behind leaders Ulster while third-placed Llanelli face the Blues on Saturday at the Arms Park.
The Welsh side recovered from their 48-20 hammering at the hands of Glasgow last time out with a resolute defensive display in the second period.
It was a very tight contest with the opening points only coming in the 27th minute with Lawlor opening penalty.
The winger added a second in the 33rd minuite after Dragons' tight-head was Chris Anthony sin-binned.
But the Dragons were rewarded with some attacking play in first-half injury-time.
Munster scrum-half Frankie Murphy's pass was intercepted by Montgomery in the Dragons' half, and Craig Warlow's half break created the overlap for Marinos.
Montgomery converted for a 7-6 half-time lead and then added three further second-half penalties to deny Munster a bonus point and inflict their fourth straight loss in the competition.
Munster (6) Pens: Lawlor 2
Dragons (16) Try: Marinos Con: Montgomery Pen: Montgomery 3
Munster: Cullen; Payne; Kelly, Holland, Lawlor; O'Sullivan, Murphy; Horan, Sheahan, McIlwham, Pusey, Hayes, Keogh, Wallace, Williams (capt).
Replacements: Flannery, Roche, Halvey, Malone, Reddan, Jones-Hughes / Matthews, McPhail.
Dragons: Tuipulotu; Brew; Montgomery, Marinos (capt), Luscombe; Warlow, Baber; Black, Jones, Anthony, Sidoli, Edwards, Beattie, Forster, Skinstad.
Replacements: Crawford, Snow, Charteris, Oakley, Jones, Winn, Breeze.
Referee: Graeme Hannah (Scotland).