Dragons scrum-half Rhodri Williams’ double proved in vain as the Welsh side failed to take any points back home.
Glasgow got off to a sloppy start when Williams intercepted Jamie Dobie’s pass to Pete Horne and scampered 80 yards to score inside the first minute.
Sam Davies landed the conversion into the howling gale and for the next 10 minutes, Glasgow played error-strewn stuff until a wind-assisted Dobie clearance took them 80 yards downfield.
Tim Swinson barged over with Horne’s conversion levelling the score. A bust-up between props Ollie Kebble and Lloyd Fairbrother earned them both yellow cards but it was Glasgow who took advantage.
They had slightly the better of the driving contest and hooker Stewart celebrated his return to a starting slot when he finished off a spell of intense pressure.
The half-hour mark was good and bad for the home side, with talisman Leone Nakawara limping off just before Horne slotted a penalty.
Both sides upped the tempo in the final 10 minutes of the first half with Huw Taylor and Harri Keddie making inroads into Glasgow’s 22 but the defence held firm to take a 17-7 lead in at the break.
Dragons earned early field position on the Glasgow 22-metre line but an astonishing break by Dobie out of a crowd of tacklers took play to the Dragons’ 22.
The line-out produced a 20-metre unstoppable drive by Glasgow and again it was Stewart who emerged with the ball to claim the try.
Horne’s kick was wide and the next try meant either Dragons were back in the game or a bonus-try point for Glasgow and the home side looked the more likely.
Great work by Chris Fusaro and Rob Harley took play back into the Dragons’ territory but it was Dragons who burst back into the game as Sam Davies split the Glasgow defence as hooker Ellis Shipp was the link and Williams raced in for his second try.
Dragons looked to pull something out of the game and when Joe Davies charged down Nick Frisby’s kick for Ashton Hewitt to snaffle a try.
Davis’ conversion would have made it a one-score game but he missed and Glasgow added gloss to the scoreline with a try in the final play from Mesu Delokoto.
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