Glasgow took an early lead through a try from Niko Matawalu but Dragons - despite having Pat Leach, Matt Pewtner and Netani Talei all spending time in the sin-bin at various points - proved to be the more cohesive side, well marshalled by scrum-half Richie Rees.
Number 10 Jason Tovey scored a try in the first half and Will Harries in the second, with Tovey kicking both conversions, two penalties and a late drop-goal en route to a convincing triumph.
It was a very disappointing night for a rusty Glasgow side that passed up a chance to go back to the top of the RaboDirect PRO12 standings.
The game kicked off in decent - though cold - conditions, and with 38 seconds on the clock one of the fastest ever sin-binnings occurred, as Dragons centre Pat Leach spear-tackled home flanker Rob Harley and received his temporary marching orders from referee Peter Fitzgibbon.
Glasgow wasted no time in making use of the numerical advantage. Centre DTH Van Der Merwe found a bit of space and the move ended with winger Matawalu touching down, but Henry Pyrgos was unable to convert.
However, before the Dragons had visited the Glasgow 22, they were ahead. Penalties, awarded at a lineout and then a scrum, were driven home by Tovey on seven and 11 minutes.
In the middle of the half, Glasgow had possession and territory but failed to provide any fluency. A penalty attempt in the 20th minute by Scott Wight hit a post.
Dragons were looking the more composed side but their try six minutes before the interval came gift-wrapped from Matawalu, who turned from hero to villain.
Having apparently dealt with a kick that ran close to his goal-line, he tried to be too clever, lost possession and the alert Tovey dived through would-be tacklers to touch down, and proceeded to convert to get his team 13-5 up at the break.
A quick response from Glasgow saw them claw back a penalty from Pyrgos five minutes into the second half.
On 53 minutes referee Fitzgibbon produced his second yellow card, this time showing it to left winger Pewtner. Seven minutes later number eight Talei followed him.
However, in between, Dragons had moved further ahead, winger Harries touching down a cross kick by Tovey, who added the extras.
Glasgow threw everything at Dragons in the closing stages but a mixture of their own errors and stalwart defence meant it was the visitors' night.