The Scottish side kept their error count low and took most of their scoring chances, with Duncan Weir kicking 14 points from five successful attempts on goal.
Canadian winger DTH van der Merwe scored an early try to celebrate the birth of his daughter Nola Grace this week, with Weir's accurate kicking boosting Glasgow's lead to 13-3 by half-time.
Dan Parks and Weir swapped penalties in the third quarter before Parks' replacement Craig Ronaldson landed a brace of kicks during a promising Connacht spell.
But the league's bottom side struggled to get their attack going and had flanker Mata Fifita sin-binned for a late tackle, leaving Weir to sew up the result for Glasgow.
The Warriors had the first use of a strong wind and hit the front in the fifth minute when Van der Merwe crashed over from close range, rewarding his forwards for a good early spell of possession.
Weir converted and the Scots' driving play and crisp handling at close quarters kept them in the ascendancy.
Mark Bennett tried his luck from a penalty in his own half but dropped it short.
Parks, who lost his half-back partner Kieran Marmion to a hip injury, responded with a penalty from in front of the posts, but indiscipline and handling errors were beginning to cost Connacht as Weir extended Glasgow's advantage.
A peach of a strike from the 22-year-old fly-half, five metres in from the touchline, was followed by a close-in kick in the 33rd minute as Connacht fell 10 points behind.
Pat Lam's side improved approaching half-time, a Park penalty attempt falling short into the wind before the home forwards went close to forcing their way over the try-line.
Kiwi flanker Jake Heenan, one of Connacht's best performers so far this season, set the standard again in terms of work-rate but Glasgow looked comfortable on the resumption.
Parks misjudged the wind when drawing a left-sided penalty wide before he right-footed home from much closer in with 51 minutes gone.
However, Connacht coughed up a cheap three points in return, the Warriors lifting their game and attacking in the 22 with Weir nailing the resulting kick.
Ronaldson fired home an inviting first penalty on his introduction, rewarding the home pack for a lineout steal and strong scrum, and Glasgow were beginning to look frustrated that they had failed to shake Connacht off their tails.
Ronaldson missed a difficult second shot at the uprights but it was his brilliant break which injected some much-needed pace into the Connacht attack entering the final quarter hour.
Although a scrum penalty on the 22 allowed Ronaldson to close the gap to four points, Glasgow were rock solid on restarts and an angled run from Van der Merwe almost led to a second try.
Niko Matawalu was then floored by a late challenge from Fifita as he launched a kick chase and there was no way back for 14-man Connacht, with a 79th-minute penalty from Weir sealing a seven-point Glasgow success.
- Sports & Recreation
- Duncan Weir