Warriors shook off a major dose of the jitters to kick-start their Pro12 glory quest but at a high cost.
Gregor Townsend’s troops produced a stirring fightback when it mattered after being under the Ospreys cosh for the bulk of the Liberty Stadium contest.
However, their preparations for the back-to-back Euro clashes with Cardiff look set to be rocked by a serious-looking injury to experienced paceman Sean Lamont.
The Scotland veteran was stretchered off with lower-leg damage after landing awkwardly in a challenge in the wake of helping to create a vital try.
Glasgow’s early dominance was rewarded when Ruaridh Jackson slotted a 35-metre penalty. But the Warriors found themselves adrift in the 15th minute.
Their first line-out mistake created the time and space for winger Jeff Hassler to speed across on the overlap. Jackson repaired some of the damage with another penalty from dead in front.
But Sam Davies calmly clipped over a drop-goal for the hosts and bagged another three points, taking advantage of the Warriors’ mounting tally of bungles.
Ospreys widened the margin to 10 via the boot of Davies from close quarters.
Then came the moment the Warriors earned fresh hope with a score that stemmed from beside their own posts, with Chris Cusiter scuttling over after being set up by Lamont.
Seconds later, Lamont was carted off in agony, Stuart Hogg then narrowed the leeway to just two on the hour-mark, followed by a tries by Dan van der Merwe and Ryan Wilson, both goaled by Jackson.