|Dragons (3) 13|
|Pens: O'Brien 2|
|Glasgow (13) 33|
|Tries: Dunbar, Strauss, Seymour, Swinson|
|Cons: Weir 2|
|Pens: Weir 2, Hogg|
Glasgow Warriors were down to 14 men for all but a minute of the second half after flanker Tyrone Holmes was sent off, but still left Newport with a bonus-point win.
Alex Dunbar went over before Holmes' red card for stamping and Josh Strauss, Tommy Seymour and Tim Swinson also touched down against disjointed hosts.
Dragons' first try by Richie Rees - after Strauss' score - gave them hope.
But it was not enough against last season's runners-up.
The outcome underlined the gulf between the Celtic rivals as Gregor Townsend's Glasgow departed with a third successive competition win, while Lyn Jones' Dragons have yet to triumph this season.
The hosts were so badly hit by back-division injuries that centre Ben John went straight into the starting line-up after arriving on loan from Ospreys.
The visitors quickly took advantage of the unfamiliarity in the home midfield when centre Dunbar scythed through for the opening try.
Weir converted the fourth-minute score and added a penalty before the opening quarter was up.
Last season's Wales Under-20 fly-half Angus O'Brien replied with Dragons' only points of the opening period.
But Weir restored Glasgow's 10-point lead soon after and the scores remained 13-3 to the visitors until the break.
Dragons' chances of causing upset seemed to receive a significant boost when Holmes was banished for stamping on home hooker T Rhys Thomas soon after the break.
Holmes was sent off on the advice of video referee Tom Hayes, while Thomas could have been sin-binned for killing possession at the ruck at which he received punishment from the Glasgow forward's boot.
Dragons were then rewarded for sustained pressure by O'Brien's second penalty.
But that good work was undone when obstruction in attack allowed Weir to kick his third penalty.
And even with only 14 men, the visitors looked more dangerous and cohesive in attack as their momentum built until lock and skipper Strauss powered over. Weir again converted.
And although scrum-half Rees' close-range dart after two powerful bursts by home captain and lock Rynard Landman gave Dragons hope, Glasgow were simply too strong.
A sweeping move sent wing Seymour past three defenders on the right in the 69th minute as Glasgow again defied their numerical disadvantage.
And the visitors' superiority was underlined when lock Swinson powered over in the dying minutes to secure the bonus point.
Dragons: Hallam Amos, Tom Prydie, Ben John, Dorian Jones, Aled Brew, Angus O'Brien, Richie Rees; Boris Stankovich, T Rhys Thomas, Lloyd Fairbrother, Cory Hill, Rynard Landman (capt), Lewis Evans, Taulupe Faletau, Nic Cudd.
Replacements: Elliot Dee, Hugh Gustafson, Dan Way, Matthew Screech, Andy Powell, Luc Jones, Ashton Hewitt, Barney Nightingale.
Glasgow: Stuart Hogg, Tommy Seymour, Alex Dunbar, Peter Horne, DTH van der Merwe, Duncan Weir, Niko Matawalu, Gordon Reid, Kevin Bryce, Euan Murray, Tim Swinson, Leone Nakarawa, Rob Harley, Tyrone Holmes, Josh Strauss (capt).
Replacements: Fraser Brown, Alex Allan, Rossouw de Klerk. James Eddie, Chris Fusaro, Adam Ashe, Henry Pyrgos, Lee Jones.
Referee: David Wilkinson (IRFU)