A resilient Connacht made Glasgow work hard before achieving their first win with a 27-17 result in the RaboDirect PRO12 at their new Scotstoun home.
Glasgow were the more enterprising side and significantly each of their four tries, 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 had a second-half brace, while Scott Wight converted two of the tries and kicked a penalty.
However, Glasgow could never shake off Connacht, who had a first-half try from Mark McRea and one in the second period from 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 Glasgow ran back a Connacht kick in exciting fashion. Again on the right wing, Lamont fed full-back Murchie who raced 40 metres for the try 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 touchdown under the bar.
Matt Jarvis converted and then in the last act of the first half kicked a long penalty to cut Glasgow's first half 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 broke up field to score under the posts for Wight to convert.
However, when Glasgow failed to deal with the restart, Connacht prop Nathan White delighted in picking up and scoring for Jarvis' conversion to narrow the Glasgow lead to two points.
It was seven minutes from time before Glasgow made sure of the win.
Substitute number 10 Ruaridh Jackson worked a bit of magic, Murchie kept the move going and Van Der Merwe touched down.