After a raft of substitutions, Glasgow wasted little time in mounting a second half comeback. Lock Richie Gray grabbed a try after just five minutes, and a long range penalty from fly-half Ruaridh Jackson pared the gap back to 12 points.
But the Dragons, who lost Jason Tovey to an early injury, stood firm to win for the first time in four Magners League games.
Giant lock Luke Charteris led the men of Gwent once again, with coach Paul Turner making six changes to the team that lost 40-17 to Connacht last weekend.
Canadian full-back Matt Evans, centre Pat Leach, scrum half Wayne Evans, prop Nigel Hall and back rowers Danny Lydiate and Gavin Thomas were drafted in.
Hampered by a rib injury, regular captain Tom Willis failed a fitness test, while Tovey made the switch from full-back to fly-half.
Dubbing this 'a huge game' and a chance to improve his side's away form, Glasgow boss Sean Lineen gave new signing, Canadian flanker Chauncey O'Toole, an immediate chance to impress.
Ed Kalman started in the front row in place of Moray Low, in the second change to the Warriors side that came from behind to beat Leinster in their league opener.
Tovey kicked the Dragons into a seventh minute lead before a hamstring problem forced him off, with Matthew Jones replacing him.
Number 10 Ruaridh Jackson made it 3-3 with his first penalty effort, before the Dragons enjoyed a dominant spell during which they racked up 20 points.
Jones kicked the first of two penalty attempts, following a brace of blockbusting tackles from Leach and Lydiate.
The game's opening try arrived in the 22nd minute, the Dragons showing their intent by kicking a penalty to touch and setting up a surging maul from the lineout. Sowden-Taylor grounded the ball with Glasgow failing to contain the drive, and Jones converted for 13-3.
The Warriors were in further trouble when their lock Tom Ryder was sin-binned for slowing up ruck ball, on the half hour, and a second Dragons try duly followed.
Brew finished off a tidy attack down the left, squeezing past two defenders to dot down from close range. Jones added the extras and another penalty to put 17 points between the sides as the half came to a close.
The Warriors hit back with a spritely start to the second half. They worked the ball out to the left for number 8 Richie Vernon, one of their best players on the day, to thunder through a gap. He drew full-back Evans and passed back inside for the supporting Richie Gray to power away from two despairing tackles for an unconverted try.
Some great footwork from replacement Hefin O'Hare got the Scottish side back on the offensive, but just when another try looked on the cards, play was called back for a forward pass from team captain Bernardo Stortoni.
Glasgow were beginning to dictate the play and a superb 55-metre penalty from Jackson kept the scoreboard ticking.
But the Dragons were able to hold them at arm's length in the closing 20 minutes. There were a couple of flashpoints, and with 13 minutes remaining, Glasgow winger DTH van der Merwe was sent to the sin-bin for his part in an off-the-ball clash with Brew.
Scoring chances were few and far between, as the Dragons deservedly took the match points and Glasgow were left ruing a poor first half showing. The result marked the Warriors' first defeat in six Magners League regular season games.
|Newport Gwent Dragons Score Card|
|Glasgow Warriors Score Card|