Ospreys full-back Adrian Cashmore got the Welsh side off to a perfect start as they went on to win in Glasgow.
Glasgow: (0) 8
Ospreys: (10) 22
Pens: Cashmore, Connor (3)
Drop Goal: Connor
The visitors opened up a 10-point lead in the first seven minutes thanks to a penalty and a try from Cashmore, which he duly converted.
Glasgow prop Kevin Tkachuk responded with a try after the interval and Colin Gregor knocked over a penalty.
But Shaun Connor slotted a neat drop-goal and added a late hat-trick of penalties to seal the victory.
The defeat is a big blow for Glasgow's hopes of claiming a Heineken Cup berth next season as they were never able to recover from a disastrous start.
Cashmore began a personal purple patch by slotting the kick after Fergus Thomson was penalised for smothering the ball.
The full-back then dived over near the left touchline after the visitors switched possession from a scrum on the opposite side, and converted his own try.
Hopes of a Glasgow comeback were dashed by a string of unforced errors and they could have been further behind but for Cashmore missing a straightforward penalty.
Gregor missed two long-distance penalties. His first dipped onto the crossbar while the next had enough distance but drifted wide.
However, the Warriors' luck changed on the restart when good lineout possession set up a ferocious drive towards the line and Tkachuk plunged over for the try - Gregor hit the woodwork for a second time when his conversion rebounded from an upright.
The Ospreys hit back with Connor's drop-goal, Gregor responded with a penalty in front of the posts, but Connor added three more penalties as Glasgow lost their discipline.
Glasgow captain Jon Petrie:
"To go from a performance like last week's game against Bourgoin to this is abysmal and I make no apologies in saying that.
"Questions must be asked of us as we've proved the team can perform well so it's not a physical problem.
"It must be a mental attitude thing that makes us such a Jekyll and Hyde team as it looked like some of the guys' heads just weren't in it tonight.
"We just didn't play our style of rugby and I'm really hacked off about it.
"We obviously have to take a hard look at what happened out there and make sure we do our best so it doesn't happen again."
Ospreys coach Lyn Jones:
"As usual Glasgow played a positive game, but our defence held firm and kept them out - that's a big plus for us.
"Our performances have been improving steadily, now we have to maintain that and get our injured players back on the paddock.
"Huw Bennett and Sonny Parker went off injured against Glasgow, but we're not sure how bad they are yet - it doesn't get any easier."
Glasgow: Lamont, O'Hare, Craig, Morrison, Roberts, McGregor, Beveridge, Tckachuk, Murray, Hayter, Turner, Swindall, Beattie, Barclay, Petrie.
Replacements: Staniforth for O'Hare (50), Cairns for Craig (25), Pinder for Beveridge (58), Harrison for Murray (61), Hamilton for Hayter (50). Not Used: Milligan, Eddie.
Ospreys: Cashmore, Terblanche, S. Parker, A. Bishop, Mustoe, Connor, Spice, Griffiths, Evans, Millward, Wyn Jones, Bateman, Dillon, Fater, Tandy.
Replacements: Karauna for S. Parker (73), Beech for Tandy (75). Not Used: Hibbard, Rees, Newman, Roberts, M. Jones.
Referee: David Hopkins (Wales).