Warriors snatched their first win in gusty Galway for six years but their efforts were not enough to earn a home Pro12 semi-final.
The success will, however, have provided a dose of confidence as they prepare instead for a battle with European giants Leinster in Dublin.
Connacht made Gregor Townsend’s troops toil but new Lions Stuart Hogg and Sean Maitland came off the bench to snuff out their challenge.
Glasgow had enjoyed an instant reprieve when former club hero Dan Parks sent a straight penalty past the far post.
And on 11 minutes Warriors stand-off Ruaridh Jackson edged them in front. His 35-metre drop-goal bid struck the upright but his luck was in as the ball rebounded over the bar.
Winger Dan van der Merwe then pounced for a trademark try. There were plenty of markers in his way as he grabbed the ball 25 metres out but he spun off the last attempted tackle by Parks and dived over.
Jackson’s conversion from the widest possible angle was spot on. There was a setback when prop Jon Welsh was yellow-carded for careless footwork.
Glasgow continued to hold the upper hand and Van der Merwe should have scored again on the half-hour mark. He had a clear run to the line but the pass from Jackson was sloppy and the Canadian could not scoop it up from the sodden turf.
Sub Hogg banged over a three-pointer to ease the jitters. And he was back in the spotlight to convert a brilliant try in the corner by Maitland.