The reigning Guinness PRO12 champions took away a bonus point win against the Welsh Region thanks to a last minute try from replacement Simone Favaro.
The Dragons began the match well but it was Glasgow who struck first with a penalty kick from flyhafl Rory Clegg.
Jason Tovey replied with his own penalty but he unfortunately missed. However the Dragons went over the whitewash and it was skipper Rynard Landman who claimed the spoils, with Tovey converting.
With the Dragons being penalised for being offside, the Warriors kicked tot eh corner which led to a converted try. However the home side were penalised just before the break with a yellow card and on the final whistle Tovey evened the scores up.
Half time: Glasgow Warriors 10 – Newport Gwent Dragons10
After 15 minutes for second half play Glasgow Warriors came up trumps with a converted try.
Flanker Simone Favaro was sinbinned later on the match and the Dragons went to the corner, unfortunately the driving lineout was stopped short.
It was Glasgow Warriors who added the next set of points on the scoreboard, after a lucky bounce sent Junior Bulumakau over, making the score 22-10.
After a strong carry from replacement hooker Rhys Buckley, wing Ashton Hewitt was helped over the try line by Tovey. The flyhalf hit the sticks with his kick. However in the dying minutes the home side struck from a driving line out to claim a bonus point win at Scotstoun.