Glasgow eased to their fourth win in five outings at the Sportsground on Friday.
Connacht had hoped to finish their home campaign in style, but after John Barclay's try inside two minutes Glasgow built up a 23-8 half-time lead which they did not let slip after the break.
But the Irish responded well with a stunning Paul Warwick try that he began inside his own 22. But Dan Parks replied with two penalties and a drop-goal and with Rory Lamont's try Glasgow were out of sight.
Conor McPhillips scored after the break to bring the hosts back to a seven-point deficit.
But substitute Hefin O'Hare, and prop Keith Tkachuk added tries for the Scots, and a late penalty try was mere consolation for Connacht.