James Arlidge, Jason Tovey, Aled Brew and Ashley Smith all crossed for the home side in a thrilling contest at Rodney Parade.
Dan Parks kept Glasgow in the hunt with 14 points but even the Scotland fly-half could stop the Dragons erupting as Paul Turner’s men move into second place in the table.
The Dragons held the Indian sign over their Scottish counterparts following seven wins from their last ten meetings.
And Turner’s men got off to the perfect start in a bid to make it eight through Arlidge’s second minute touchdown.
Fly-half Arlidge pounced on Mark McMillan’s slow service to intercept and coast home from 45 metres.
Arlidge converted but the Gwent region pressed the self-destruct button from the re-start with Parks handed the chance to boot Glasgow back to within four points.
But the Dragons were back in command in the 14th minute when Tom Riley cut through a static defence and after supplying fellow centre Smith, with Tovey on hand for a converted try under the posts.
Parks hammered a long-range penalty from his own half before the Dragons grabbed their third try at the end of a frantic opening quarter.
Arlidge’s re-start was fumbled by the visitors’ and wing Brew was given a soft score in the right corner.
But Glasgow put themselves back into contention on 29 minutes when wing Colin Shaw finished off a line-out move from close-range.
Left-footed full-back Tovey took over the kicking duties from Arlidge on the far touchline and with the angle in his favour, slotted an excellent penalty attempt through the posts five minutes before the break after Glasgow failed to roll away.
The Dragons could and should have held a bigger advantage but a combined of ill-discipline and the excellent form of Parks, the Magners League leading points-scorer, kept Glasgow afloat.
Parks added his third penalty to reduce the deficit to 22-16 before Arlidge dragged his attempt wide of the posts on the stroke of half-time.
Parks notched a fourth successful kick but Tovey shut the close with his second penalty two minutes later.
And the Dragons were back on the front foot with Richard Fussell making the incisive break before Arlidge’s delightful grubber kick allowed Smith to pounce for the bonus point.
