A resilient Connacht made Glasgow work hard before the Scottish side achieved their first win at their new Scotstoun home.
Glasgow were the more enterprising side and significantly each of their four tries in the 27-17 win, which secured a bonus point, fell to a back-three player.
Right-winger Sean Lamont grabbed the first and then sent full-back Pete Murchie in for the second.
Canadian international DTH van der Merwe touched down twice in the second half, while Scott Wight converted two of the tries and kicked a penalty.
However, Glasgow failed to shake off their Irish visitors, who had scored a first-half try through Mark McRea and touched down in the second period via Nathan White, with Nathan Jarvis converting both and putting over a penalty.
The big news for the home fans in the crowd of 3,646 before kick-off was that international stand-off Duncan Weir had injured a knee in training, meaning the inexperienced Wight got a rare start.
And fortune was with him as early as the first minute when Connacht dived in at a ruck, giving Wight a straightforward penalty to steer home.
Wight missed with a long penalty four minutes later but the game sprang to life at the end of the first quarter. Connacht full-back Gavin Duffy spilled a steepling kick from Wight and the home side pounced on the ball with James Eddie sending Sean Lamont in at the right corner for an unconverted try.
Seven minutes later, Lamont fed full-back Murchie, who raced 40 metres for the try, which was converted by Wight.
Leading 15-0, Glasgow looked comfortable, but Connacht hit back in the closing stages of the first half. A clever sidefoot from sub Jack Carty allowed left winger McCrea to touch down under the bar.
Matt Jarvis converted and then kicked a long penalty to cut Glasgow's first-period lead to 15-10.
When Connacht failed again to deal with a towering Wight kick on 48 minutes, the ball broke to Van Der Merwe who scored under the posts for Wight to convert.
Glasgow failed to deal with the restart, and Connacht prop Nathan White scored with Jarvis converting to narrow the Glasgow lead to two points.
Glasgow made sure of the win seven minutes from time, when substitute 10 Ruaridh Jackson worked free Murchie and Van Der Merwe touched down.