The game began badly for the hosts and they were soon three points down thanks to an early penalty from the Ospreys Adrian Cashmore.
The fullback also crossed for the first try of the game soon after when he got on the end of a sweeping move before adding the extras himself.
The Warriors did not help themselves by throwing away an opportunity when Rory Lamont kicked too far ahead with Mike Roberts the better option.
Cashmore then failed to heap woe on the hosts when he missed a straightforward penalty after an infringement by Gregor Hayter.
The signs were there to suggest it was not to be the Warriors' day when stand-off Colin Gregor narrowly missed two long-distance penalties before the break.
To their credit, Hugh Campbell's side rallied in the second half and agood lineout and drive saw prop Kevin Tkachuk plunge over for the try. Gregor's conversion however hit woodwork.
The Ospreys took the initiative once again when Shaun Connor slotted a neat drop-goal but a penalty from Gregor gave the Warriors hope once more.
But three more penalties from Connor ensured the Ospreys held on for the win.
|Glasgow Warriors Score Card|
|Ospreys Score Card|