Glasgow asked questions of the visitors defence, but Connacht soaked up early pressure before taking control and wrapping up the issue by the interval.
Glasgow lock Joe Beardshaw got in for a try after 20 minutes after Mark McHugh had opened Connacht¹s account with a long range penalty two minutes earlier.
Mike Walls responded with an excellent breakaway try which McHugh converted before the full-back landed another penalty.
Connacht¹s second try came after 41 minutes when Matt Mostyn finished in style. McHugh added the points and then added another penalty to leave it 23-5 at the break.
Inevitably, Glasgow hit back after the break but struggled to break down a superb Connacht defence and indeed it was Connacht who increased their lead when Conor McPhillips scored his third try of the campaign after 56 minutes.
McHugh made it six kicks from six with the conversion to put Connacht in a commanding 30-5 lead.
After that Connacht, who had earlier defeated Edinburgh and The Borders, had to withstand strong Glasgow pressure but while Cameron Mather and Lee Harrison got in for tries, the issue was long decided.