Bernardo Stortoni, DTH van der Merwe and Colin Shaw scored the Warriors tries that helped seal the win with a Will Harries try and 14 points from Jason Tovey's boot the Dragons scorers.
The win puts Edinburgh third in the Magners League table with 39 points - seven shy of leaders Leinster - while the Welsh region are sixth on 33.
And despite missing out on a four-try bonus point Glasgow boss Sean Lineen was a happy man.
"I'm absolutely delighted," said Lineen.
"We wanted the [four try] bonus point but considering we were playing against a team who have won here five times in a row, I'll take the win.
"The guys just had to go out there and dig out a win.
"I though Colin Shaw showed a great deal of maturity. Bernie [Stortoni] was great under the high ball as usual and DTH [van der Merwe] was outstanding.
"It was good to get the win because we need to get momentum back into our game."