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.
|Connacht Rugby Score Card|
|Glasgow Warriors Score Card|