The Dragons had enjoyed a 16-9 advantage in the first half but they had to settle for a losing bonus point as Weir ruthlessly exploited the slightest infringement with his deadeye kicking.
Having signed a contract slightly earlier in the week, Weir entered the record books by equalling the RaboDirect PRO12 and Glasgow record of eight successful penalties in a single match.
The game began with both teams testing each other out with a series of booming kicks and it was Glasgow who drew first blood via a Weir penalty.
However the Dragons' response was swift as it was efficient with a wonderfully crafted try on six minutes.
Good lineout ball allowed the back division to purr into life with Matthew Jones and Mike Poole combining to put full-back Thomas over. Jones converted the score and soon added a penalty to give the visitors a 10-3 lead.
However Weir quickly reduced that to a one-point deficit with two penalties, the latter after a late hit by Mike Poole on Stuart Hogg.
But Jones responded in kind by nailing two out of three penalties to take the visitors ahead by seven points.
Glasgow then enjoyed a good spell of pressure and came close to fashioning a try of their own as a brilliant run from full-back Hogg was only halted just short of the Dragons.
But in what was quickly developing into a battle of the fly-halves, Weir slotted a further penalty to give the home team a five-point deficit to overturn as they went into the break 16-12 behind.
That gap was narrowed still further to a solitary point when the otherwise impressive Jones was turned allowing Weir to slot a 44th minute penalty.
With the Firhill crowd then fully behind the Warriors, the home fans were celebrating a try as Colin Shaw smashed his way over only for the score to be ruled out for a Glasgow infringement.
Another visit to the Dragons 22 - this time via a barging run from centre Rob Dewey - did result in points for an away offence as Weir booted Glasgow into the lead for the first time since the match's early stages.
Ten minutes later, Weir made it 21-19 as he maintained his 100per cent record from the tee.
Replacement fly-half Jason Tovey responded with a penalty from his own but straight from the restart the Dragons conceded yet another penalty which Weir needed no second invitation to stroke over the resulting kick.
It was all hands to the pump in the closing stages, but despite hammering away at the Warriors tryline the Dragons could not fashion a second try as their losing away record in the RaboDirect PRO12 continues.
|Glasgow Warriors Score Card|
|Newport Gwent Dragons Score Card|