The fly-half did not miss a single kick at goal, landing all eight attempts to take his success rate for the season to over 80 per cent.
The Dragons have not won on Scottish for nearly two years, but Darren Edwards' side looked well on course to end that barren run when they led 16-12 at half-time.
Indeed, the Dragons scored the only try of the night on six minutes to get them off to a fantastic start. Good line-out possession was secured before the ball went through several pairs of hands in the backs in a move that eventually led to full-back Martyn Thomas diving over. Matthew Jones converted and added two penalties in the first half to Weir's four kicks at goal.
Weir added three more penalties as Glasgow got their noses in front before Jason Tovey pulled three points back for the visitors to make it 21-19 with 14 minutes to go.
But the Dragons, plucky losers on a number of occasions already this season, were unable to force another score and it was left to Weir to punish yet another indiscretion and bring the game to a close with his eighth and final penalty of the night.
Weir has now been successful with his last 13 kicks at goal.