After last week's first league defeat of the season to Munster, Gregor Townsend's troops required a rapid return to the winner's rostrum and a try from the flying winger after six minutes gave their coach a huge boost after that difficult loss.
"We know how tough Connacht are here and we were missing a number of players on Scotland duty, but the players who came in did a fantastic job for us," said Townsend.
"We now have the two-week break for the internationals to look forward to, but we'll go back and we'll prepare now with a bit of confidence on the back of that win. We didn't have enough possession in the first half but the effort in the second half was excellent I thought."
With the gale at their backs it was no surprise to see Warriors camp firmly inside the Connacht half for the opening action. And after playing to the conditions superbly with five minutes of sustained phases, van der Merwe stretched across the fallen Connacht defenders to touch down his side's opening try.
The effect of the conditions on the match cannot be exaggerated. In the ninth minute, former Warrior Dan Parks was forced to kick to the corner when his side were awarded a normally kickable penalty from in front of the posts, and when he finally took on the posts it was from less than 20 metres after Glasgow wandered across the offside line.
Parks drilled the kick over to reduce the Warriors lead to four points, but Weir restored that cushion in the 23rd minute with another fine kick from the right touchline.
At that stage Warriors fully deserved their advantage. They were lording the breakdown, and seemed to be gaining massive yards after the tackle.
And in the 33rd minute their pressure told when Neil Patterson penalised the Connacht front row under their own posts, and once more Weir made no mistake for a 13-3 lead.
In the closing minutes Connacht came agonisingly close to their first try of the game, with Rodney Ah You and Jake Heenan both coming close to crossing the whitewash, but Glasgow held solid for a 13-3 lead at the break.
Six minutes after the restart Parks missed again from the left after another Glasgow offside infringement, but five minutes later he made amends after another offside call, this time a far simpler kick from under the posts.
After Parks departed the action, Craig Ronaldson assumed the kicking duties for the home side and he made inroads on the scoreboard in the 61st minute with a penalty awarded after Warriors wheeled the scrum.
He missed the target two minutes later with a more difficult kick, but then put just four points between the sides with 11 minutes remaining after another penalty won at the scrum by the home side, but Weir's fourth penalty late on sealed a hard fought win for Warriors.
Connacht: G Duffy; F Carr, D Leader, D McSharry, M Healy; D Parks (C Ronaldson 58), K Marmion (P O'Donohoe 27); B Wilkinson ((JP Cooney 76), J Harris-Wright (D Heffernan 55), R Ah You (N White 56); M Swift (c) (M Kearney 56), C Clarke; M Fifita, J Heenan, G Naoupu (A Muldowney 67).
Glasgow Warriors: N Matawalu; B McGuigan, M Bennett, G Ascarate, DTH Van Der Merwe; D Weir, C Cusiter (c); G Reid (M Low 56), D Hall (F Brown 62), J Welsh (E Kalman 47); J Gray (L Nakarawa 67), T Ryder; R Harley, C Fusaro (T Holmes 62), J Strauss (R Vernon 70).
Referee: N Paterson (SRU)