Connacht didn’t take advantage of their first-half chances and second-half tries from Ali Price and Lee Jones sealed the win for Glasgow.
This Conference A clash was dogged by the weather from the outset and in a 40 minutes dominated by the wind and rain Connacht held a 9-3 interval lead.
Jack Carty slotted three penalties in the first half to cancel out a solo effort from Peter Horne.
But after the break Glasgow opened up and ensured they left with the win again.
Connacht made the perfect start and they put pressure on the Glasgow defence right from the off. They forced a mistake and after the visitors didn’t roll away on time Carty kicked a second-minute penalty.
The conditions were dreadful for rugby and after Glasgow worked their way back up the field Horne sensationally drove a relatively easy kick short when it was caught by the wind.
Glasgow re-grouped, though, and Horne made amends with his second attempt from the opposite side of the pitch. That made it 3-3 after 12 minutes and Connacht debutant openside flanker Jarrad Butler went close to a try but his attempt was ruled out by TMO Peter Fitzgibbon.
With the wind at their backs and camped deep in Glasgow territory Connacht needed to leave with a score and that came by way of a second Carty penalty.
Connacht were facing a Glasgow side minus their much vaunted British & Irish Lions contingent but they couldn’t make use of the treacherous conditions. And it all stemmed from the scrum where the hosts were being demolished on every occasion.
Connacht held their line with a vital penalty when Glasgow strayed off-side in the 25th minute, but handling errors from both sides meant neither could really gain the ascendency.
Cian Kelleher almost got through shortly before the half-hour but despite some brilliant work from Bundee Aki in the build-up he kicked the ball directly out of play.
Even though the error-count rose both teams continued to throw the ball around and neither chose to kick deep for territory.
It was to Connacht’s detriment and they missed out on numerous try-scoring chances before finally Scott Cummings conceded a late penalty and Carty tagged on his third successful kick.
Glasgow had the wind to come in the second-half, they only trailed 9-3 at the break and the conditions worsened on the resumption.
Glasgow quickly cut back the Connacht lead. It was down to just three after six minutes when Horne kicked his second penalty.
The Connacht defence looked shaky and Horne found space behind the cover before he fed Price for the crucial Glasgow try in the 49th minute.
It was a massive moment in the game and Horne’s conversion compounded the error for the hosts as Glasgow led 13-9.
Connacht could have given up at that stage, but they pulled through a difficult period to earn a penalty in the Galway 22, and Carty kicked his fourth to bring the deficit back to the minimum.
There were 19 minutes remaining but Connacht could not find a way back.
Jones effectively settled it with his try in the corner and Glasgow held on despite a desperate multi-phase handling effort by the home side in search of the converted try they needed to snatch victory.