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Tom Habberfield's counter-attack try was the only score of the first half.
Warriors lock Brian Alainu'uese was sent off for a charge to the head of Rory Thornton six minutes after the break.
Tom O'Flaherty and Dan Baker touched down soon after, with Ashley Beck grabbing the bonus-point try before Junior Bulumakau's consolation.
Ospreys move into second place in the Pro12 behind Leinster, while Glasgow drop out of the play-off spots behind Munster and Scarlets.
The autumn internationals deprived these sides of a combined 26 players, with other Test regulars still injured.
Neither the patchwork nature of both line-ups, nor the polar temperatures and slick ball, did much for the fluency of the contest.
There were fleeting glimpses of first-half flair and dynamism, Ospreys fly-half Josh Matavesi puncturing the home midfield, while Alainu'uese was an industrious and effective battering ram before his evening was curtailed.
Far more precision, though, is expected from a Warriors line-out spearheaded by an ex-All Black and World Cup-winning hooker in the shape of Corey Flynn - his first four throws failed to hit their target.
Meanwhile, at scrum-half, Grayson Hart was too desperate, and too impatient, to be the fulcrum in his first start of the season, flinging several ambitious offloads that were never likely to come off.
Habberfield's try, on 37 minutes, was what is known in modern sporting parlance as a "coach-killer", a sting in the tail at the end of a turgid first half.
Lewis Wynne, the Warriors eight, lost possession in contact and neither Flynn nor Rory Clegg booted the loose ball off the pitch, nor fell to secure it.
Instead, the Ospreys scooped it up, Daf Howells punted downfield into yards of unguarded turf, captain Habberfield hacked on and plunged to touch down ahead of the desperately back-tracking Nick Grigg.
Prior to that, Ospreys had more than 60% possession and territory - and though Matavesi's conversion struck the post, Steve Tandy's side held a 5-0 half-time lead for their toil.
World Rugby has recently ordered its referees to police high tackles, charges and contacts to the head with greater stringency, given the prevalence and peril of head injuries in the game.
Six minutes after the break, Alainu'uese stooped to clear Thornton, his opposite number, from a ruck, his shoulder and cocked upper arm connecting with the head of the Osprey, sending him sprawling.
George Clancy studied the television replays before dismissing the towering lock and incensing Scotstoun.
Some may deem the sanction an affront to good, hard rucking, but under the World Rugby directive, Clancy had cause to show red.
The Ospreys duly capitalised, replacement O'Flaherty gathering Matavesi's arrowed crosskick to stretch the lead and number eight Baker powering over before the hour.
Beck swerved and slalomed home from five yards to notch the bonus-point try, Matavesi finally slotting a conversion.
Patient Glasgow phase play culminated in Bulumakau scampering in from Clegg's looping pass in the final two minutes to at least prevent a whitewash.
Glasgow Warriors coach Gregor Townsend: "I'm obviously disappointed we didn't win at home and disappointed for the crowd that turned up in big numbers.
"I was really proud of the effort. I'm proud of the way we defended in that first half, when Ospreys had a lot of possession, and proud of the way we finished the game.
"That last 15-20 minutes, when we were a man down and had a lot of young players, they did really well and it was great to see them score a try.
"We obviously have a bit of improving to do and you're not going to beat a team like Ospreys when they're playing so well when you're down to 14 men for such a long time."
Glasgow Warriors: Murchie, Lamont, Grigg, Johnson, L. Jones, Clegg, Hart, Reid, Flynn, Rae, Swinson, Alainu-uese, Wynne, Fusaro, Strauss.
Replacements: Bulumakau for L. Jones (69), Grant for Reid (55), MacArthur for Flynn (55), Nicol for Rae (59), Haupeakui for Fusaro (59), McAlpine for Strauss (74). Not Used: G. Horne, Kelly.
Sin Bin: Alainu-uese (47).
Ospreys: D. Evans, Howells, Fonotia, Beck, Grabham, J. Matavesi, Habberfield, James, Parry, Fia, Ashley, Thornton, Cracknell, Underhill, D. Baker.
Replacements: Price for Beck (74), O'Flaherty for Grabham (44), Leonard for Habberfield (67), Otten for Parry (59), R. Jones for Fia (55), Beard for Ashley (70), McCusker for Underhill (67). Not Used: G. Thomas.
Referee: George Clancy (Ireland).
Ospreys earned their first win in Glasgow since 2009 by easing aside the 14-man Warriors at Scotstoun.