Michael McCarthy (left) tackles Glasgow's Opeta Palepoi
Connacht (12) 15
Tries: Carr, Wilkinson Cons: Keatley Pen: Dunne
Glasgow (8) 8
Try: Evans Pen: Parks
Connacht crept off the bottom of the Magners League table as their rock-solid defence frustrated Glasgow Warriors into submission in Galway.
A Dan Parks penalty and Thom Evans try helped Glasgow lead 8-0 but tries from Fionn Carr and prop Brett Wilkinson left Connacht 12-8 ahead at half-time.
Glasgow failed to make the most of several scoring opportunities in the second half.
Andy Dunne's late penalty ensured the home team's first win of the campaign.
Glasgow's place-kickers Parks and Colin Gregor both had a night to forget as they were off target on several occasions.
Parks did land a second-minute penalty goal and winger Evans burst through for a fine individual try after 22 minutes.
Ireland international Gavin Duffy failed to retrieve a high kick and the alert Evans lapped up the loose ball and showed impressive pace to run in his third try in two games.
Glasgow were dominating possession but Parks' missed conversion and two sliced penalties did little for his team's confidence.
Duffy redeemed himself with a try-saving tackle and full-back Carr really announced himself with a superb 31st-minute try.
The summer signing intercepted a pass from Graeme Morrison on the Connacht 22 and ran all the way to the Glasgow line.
Ian Keatley converted and Connacht moved ahead soon afterwards when scrum-half Frank Murphy sliced through a gap and offloaded for Wilkinson to pile over.
Connacht suffered two sin-binnings in the third quarter with Colm Rigney and Duffy both carded for not rolling away in the tackle while Welsh referee David Bodilly also flashed a yellow at Glasgow centre Morrison.
Connacht's forwards, most notably the heavy-hitting Mike McCarthy, tackled their hearts out in a tense finale as Glasgow dominated possession.
Their best scoring chances fell to Gregor who twice shot wide of the uprights and it was left to replacement Dunne to guarantee victory for Connacht with an injury-time penalty.
Connacht: F Carr; L Bibo, G Duffy, K Matthews, J Hearty; I Keatley, F Murphy; B Wilkinson, S Cronin, R Morris; M McCarthy, A Farley; J Muldoon (C), R Ofisa, C Rigney. Replacements: A Flavin, R Loughney, A Browne, J O'Connor, K Campbell, A Dunne, M Deane.
Glasgow: B Stortoni; H O'Hare, M Evans, G Morrison, T Evans; D Parks, M McMillan; J Va'a, D Hall, M Low, O Palepoi, A Kellock (C), S Swindall, J Barclay, K Brown. Replacements: E Milligan, K Tkachuk, D Turner, Beattie, A Henderson, C Gregor, R Jackson.