Colin Gregor scorched away and found John Barclay inside him. The Warriors skipper was hauled down inches short but rapidly recycled ball gave Morrison space to go over, giving Jackson a straightforward conversion.
At 16-7 ahead heading towards the break, the Warriors were in control, but a rush of blood to the head gave Jones the chance to bring the margin back to six points at half-time with a simple penalty.
Jones narrowed the margin further with a penalty four minutes into the second half, but then the Dragons were under the cosh for 15 minutes, and only solid midfield defence from Thomas Cheeseman and Ashley Smith kept Max Evans in particular at bay as the Warriors started to put real pace on the game.
Argentine wing Federico Aramburu had gone close a couple of times for the Warriors and from his second dip at the line, quick ball found the Dragons way out of position and hooker Fergus Thomson went over for Glasgow's second try.
Although Jackson missed the conversion, the home side were growing in confidence.
Referee Garces was doing little to endear himself to the Glasgow crowd, but the home side looked to be running the clock down successfully.
Then a Wayne Evans kick left the hobbling Bernardo Stortoni stranded and Glasgow were penned in their own 22 until man-of-the-match Gregor got them out of trouble with a couple of fine clearances.
Jones had a last-minute attempt to salvage the bonus point but the kick fell short, a Dragons final flurry came to nothing and Glasgow had ended their losing streak.
Glasgow Warriors: Stortoni, van der Merwe, M. Evans, Morrison, Aramburu, Jackson, Gregor, Welsh, Thomson, Low, Ryder, Gray, Harley, Barclay, Vernon. Replacements: Pyrgos for Stortoni (72), Dewey for van der Merwe (78), Weir for Jackson (63), Grant for Welsh (70), Hall for Thomson (63), Kalman for Low (60), Forrester for Ryder (16). Not Used: Muldowney.
Newport-Gwent Dragons: Harries, Hughes, Cheeseman, Smith, A. Brew, M. Jones, W. Evans, Price, S. Jones, Castle, Morgan, Sidoli, Lydiate, Sowden-Taylor, Coombs. Replacements: Riley for Smith (64), Gustafson for Price (77), Burns for S. Jones (63), Hall for Castle (40), M. Evans for Lydiate (16), L. Evans for Coombs (60). Not Used: A. Jones, Leadbeater.
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