Scrum-half Niko Matawalu had shown his talent by scoring a length-of-the-pitch solo try at the end of the first half, but then with minutes to go he first threw a wild pass that missed full-back Ruaridh Jackson and then had a kick charged down from the following ruck.
That allowed Dragons replacement wing Ross Wardle to pounce on the loose ball, sidestep the despairing Matawalu and sprint over in the corner. Full-back Rhys Jones landed the touchline conversion to put his side one point clear and allow the Dragons to hold on for victory.
Glasgow outside-half Finn Russell did have a kick of more than 40 metres to restore the lead, but it fell short into the wind. Had centre Mark Bennett still been on the pitch he would have had the range to reach, but he had been substituted.
Although Glasgow were masters of their own downfall, they were also unlucky to have been put in that position by Italian referee Marius Mitrea.
Firstly Henry Pyrgos was blocked when he chipped through to the Dragons line and seemed to have a clear run for the score. Glasgow were awarded the penalty, which Russell kicked, but that should have been accompanied by a yellow card which was not forthcoming.
Then another kick through to the Dragons line was kicked loosely back. Glasgow attacked from inside the Dragons 22 only to be stopped by all the Dragons players who were miles offside. The referee might have thought the kick had been touched in flight rather than sliced, but that should have been a penalty between the posts which would have sealed Glasgow's win.
To make matters worse, the Dragons stole the ball, kicked upfield and that led to their match-winning score. "Very disappointing. To lose a game is always disappointing but to lose it in the last five or six minutes is really frustrating," admitted coach Gregor Townsend.
"I thought Niko played well, he was at the heart of a lot of our positive play. He had the kick charged down at the end but I will certainly be having a look at what happened before that.
"It was a tough kick for Finn, the wind was swirling in the second half. I thought he had a good game, got some good kicks and was one of our better defenders in the second half. To go down by a point . . . there are other things we are concerned about but not that kick."
Glasgow got off to the perfect start when Rob Harley charged down a Jason Tovey clearance and picked up the ball to score.
Russell converted and then Bennett landed a long-range penalty, before Matawalu kicked out of defence, regathered from a fortunate bounce and had the pace to go all the way.
Two more Russell penalties seemed to have done the trick, but a break by Richie Rees, the former Edinburgh scrum-half, set up lock Cory Hill for a try before the final chargedown.
Scorers. Newport Tries: O Evans, Hill, Wardle. Cons: Tovey, R Jones (2). Pen: Tovey.
Glasgow Tries: Harley, Matawalu. Cons: Russell (2). Pens: Russell (2), Bennett.
Newport Gwent Dragons R Jones; M Pewtner, P Leach, A Smith (Capt), W Harries (R Wardle 45-51, 58); J Tovey (D Jones 23), R Rees; O Evans (P Price 58), T R Thomas, D Bell (B Douglas 45), A Coombs, C Hill, J Groves (M Screech 65), N Cudd, L Evans.
Replacements: H Gustafson, I Jones, W Evans.
Glasgow Warriors R Jackson; L Jones, R Vernon, M Bennett (F Niua 65), B McGuigan (H Pyrgos 59); F Russell, N Matawalu (R Hughes 73); J Yanuyanutawa (G Reid 45), D Hall (F Brown 51), M Cusack (J Welsh 37), L Nakarawa (T Ryder 51), J Gray, R Harley, T Holmes (K Bryce 65), J Strauss (capt)
Referee Matteo Liperini (Italy)