The Dragons scored first, winger Adam Hughes crossing in his first European game and Matthew Jones added the conversion.
Ruaridh Jackson kicked two penalties to open Glasgow's account after 20 minutes and brought the scores to within a point.
The 22-year-old fly-half kicked Glasgow into the lead for the first time with another penalty just before Scotland centre Graeme Morrison crossed for a try for the home side, Jackson adding the conversion.
Jones kicked a penalty for the Dragons on the stroke of half time to reduce the deficit to 16-10.
The Welsh fly-half was on the score sheet three minutes after the break with another penalty, but that was the last score of the evening for the Dragons.
Hooker Fergus Thomson scored Glasgow's second try to give the home side victory for the first time in the opening weekend since 1997.
The Scottish side will look to continue their good start when they travel to face double European champions London Wasps next weekend, while the Dragons must regroup for the visit of the Heineken Cup Champions Toulouse to Rodney Parade.
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