Aly Muldowney's late effort, converted by Duncan Weir, brought an end to the Dragons' three-match winning run in the Magners League.
A try from lock Andrew Coombs, allied to 11 points split between goal-kickers Jason Tovey and Matthew Jones, proved to be insufficient as the Welsh region failed to hang on to their advantage.
The Dragons were searching for their first victory at Firhill Stadium since January 2009. They took an early lead after gradually working their way into scoring range.
Despite the fact that the majority of attacking movement was lateral, the visitors' persistence paid off as the home defence coughed up a penalty 15 metres from the line and fly-half Jones stroked over the kick.
His young counterpart Weir had an opportunity to cancel the deficit, but his penalty effort from the halfway line was pushed wide.
Just minutes later, Glasgow were made to pay for their fly-half's miss when Jones converted a kick from the 40 metre mark to stretch his side's advantage.
The Dragons almost added to their lead soon after, when slick handling in midfield allowed full-back William Harries to dart towards the corner. However, a last-ditch covering tackle from home winger Hefin O'Hare averted the danger.
James Eddie, along with fellow back rowers Ryan Wilson and Callum Forrester, were proving a handful around the fringes. It was their interplay which resulted in Weir having another penalty opportunity - but again he pushed his attempt wide of the uprights.
A further opportunity was missed ten minutes before half-time, Weir skewing his penalty attempt wide from an admittedly testing position tight to the touchline.
It did not take long for him to make amends after the restart however, punishing the Dragons for an infringement at the scrum with a successful penalty.
He then added another three-pointer on the 56-minute mark to level the scores after good pressure from the Glasgow forwards
The visitors found another gear. Firstly, winger Adam Hughes engineered a one on one, chipping over the towering Canadian flyer DTH Van der Merve, only to fail to gather the ball as it bobbled across the surface with the try-line at his mercy.
The score was delayed only briefly, and when Henry Pyrgos had a box kick yards from his own line charged down, the Dragons grabbed back the lead. Coombs got the touch down, with replacement fly-half Tovey added the straightforward extras.
A third successful Weir penalty kept the hosts within touching distance, but Tovey's penalty minutes from the end appeared enough to secure his side victory.
In the dying moments, referee Peter Fitzgibbon reduced the Dragons to 14 men, replacement prop Benjamin Castle sent to the sin-bin for a cynical foul at the breakdown.
The resulting play led to Weir bundling over the visitors' try-line, only to be held up. However, from the ensuing scrum, Muldowney powered over beneath the posts. It was left to Weir to coolly add the conversion and draw the sides level for the final time.
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