Tries were hard to come by at Rodney Parade but the Dragons finally managed two to Glasgows'one and had the benefit of five successful penalty kicks.
Dragons (9) 27
Tries: Forster, Thomas. Cons: Warlow. Pens: Warlow 5
Glasgow (5) 5 Tries: Lamont
Most of the early play took place in the Glasgow half but tall winger Sean Lamont caused concern when he ran through some hesitant Dragons tackling.
The attack was halted but the same player crossed for a try wide out one minute later when he took a scrummage feed from number eight Paul Dearlove.
Craig Warlow was off-target with his first penalty attempt for the home side but fired two over in quick succession to put the Dragons into the lead.
Glasgow prop Matt Proudfoot was banished to the sin-bin for the statutory 10 minutes for killing the ball on the ground and from the resultant penalty Warlow added three more points for a half-time score of 9-5.
After the break Glasgow's Daniel Parks was just off the mark with two penalty attempts, the first from inside his own half.
Steve Winn was obstructed when chasing a chip ahead and Warlow further extended the Dragons' lead with his fourth penalty.
The big crowd were given something to cheer when, following a series of scrummages on the Glasgow line, Jason Forster forced his way over in the corner for an unconverted try.
Sensing a second victory in the Celtic League was theirs for the taking, the Dragons mounted a series of powerful raids and only strong tackling and some big hits kept them from adding to their try score.
Warlow slotted his fifth penalty and with the last movement of the game replacement prop Rhys Thomas drove over for a second try. This one was converted by Warlow.
Dragons: Brett Davey; Nathan Brew, Andy Marinos (capt), Steve Winn, Ben Breeze; Craig Warlow, Gareth Baber; Adam Black, Steve Jones, Chris Anthony, Ian Gough, Peter Sidoli, Rhys Oakley, Jason Forster, Shawn Van Rensburg.
Replacements: Rhys Thomas, Paul Young, Luke Charteris, Jamie Ringer, Gerald Arasa, Lee Jarvis, Rhodri Jones.
Glasgow: Stuart Moffat, Jon Steel, Graeme Morrison, Alan Bulloch, Sean Lamont, Daniel Parks, Kenny Sinclair, Matt Proudfoot, Simon Gunn, Lee Harrison, Nathan Ross, Joe Beardshaw, Cameron Mather (capt), Paul Dearlove, Donnie Macfadyen.
Replacements: Scott Lawson, Euan Murray, Andrew Hall, Roland Reid, Mark McMillan, Calvin Howarth, David Millard.
Referee: Simon McDowell (Ireland)