Glasgow (8) 17
Tries: Morrison. Cons: Pens: Jackson 4
Connacht (13) 19
Tries: Carr. Cons: Keatley Pens: Keatley, 4
Ruaridh Jackson (centre) kicked four penalties in the Warriors' defeat
Glasgow Warriors squandered an eight-point lead as Connacht came roaring back to secure victory at Firhill.
Ruaridh Jackson struck an early penalty for the hosts and Graeme Morrison's try extended their lead.
But Connacht responded in impressive fashion with Fionn Carr surging through the Warriors defence to score.
The boot of Ian Keatley put daylight between the sides and Jackson, despite three second-half penalties, could not clinch victory as Connacht held on.
Warriors, showing four changes from the side which lost to Dragons last week, looked fired-up from the start on a clear, crisp night at Firhill and took an early lead.
Jackson struck a penalty for the Warriors to open the scoring.
Max Evans then made a break for the hosts and eight or nine phases later, Morrison powered his way over the line in the corner and the officials deemed it was a try.
But the hosts appeared to take their foot off the pedal after that impressive start and Keatley reduced the deficit with some good kicking through the posts.
Poor defensive work allowed Carr to speed away from about 45 metres out for the touchdown, and Keatley converted.
The same player kicked the visitors further ahead after the break but Connacht were penalised for not releasing and Jackson's boot reduced the deficit with a gun-barrel straight strike.
And Jackson, warming to the task, fired another penalty moments later as the home supporters roared on their spirited comeback.
Glasgow were kicking for territory and seeking Connacht mistakes as they looked to re-take the lead.
Jackson kicked Glasgow into the lead halfway through the second half, his last action of the game before being replaced by Duncan Weir.
But the boot of Keatley kicked Connacht into the lead once again with five minutes remaining.
Weir's unsuccessful drop goal in injury-time compounded the hosts' misery as Connacht secured their first away win in two years.
Glasgow Warriors coach Sean Lineen: "There's incredible frustration and disappointment. They certainly put the effort in. Connacht had a gameplan.
"They defended well and took their try; Fionn Carr is a dangerous runner. But we just stopped going forward.
"They were tackling hard and we just didn't mix it up enough. We need to look at our direction and what we're doing."
Glasgow Warriors: B Stortoni, DTH van der Merwe, M Evans, G Morrison, H O'Hare, R Jackson, C Gregor, J Welsh, F Thomson, E Kalman, T Ryder, R Gray, R Harley, J Barclay (capt), R Vernon.
Replacements: P MacArthur, K Tkachuk, M Low, A Muldowney, C Forrester, H Pyrgos, D Weir, P Murchie.
Connacht: G Duffy; T Nathan, N Ta'auso, K Matthews, F Carr; I Keatley, F Murphy (c); B Wilkinson, S Cronin, J Hagan; M Swift, B Upton; M McCarthy, R Ofisa, M McComish.
Replacements: A Flavin, R Loughney, R Sweeney, A Browne, S Conneely, C Willis, M Nikora, T O'Halloran.