Friday's Scottish opponents appeared set to inflict the Ospreys' first defeat of the new campaign until second row Wyn Jones ensured the Ospreys returned home with a deserved four points.
Fly-half Daniel Biggar kicked two penalties and a drop goal, with Kristian Phillips crossing for the first of the two Ospreys tries and James Hook converting Wyn Jones' match-winning effort.
Dan Parks became the leading scorer in Magners League history but his five missed kicks cost the home side dear.
Glasgow: Bernardo Stortoni, Lome Fa'atau, Max Evans, Graeme Morrison, Thom Evans, Dan Parks, Mark McMillan, Justin Va'a, Dougie Hall, Moray Low, Tim Barker, Alastair Kellock, Kelly Brown, John Barclay, Johnnie Beattie
Replacements: Eric Milligan, Kevin Tkachuk, Opeta Palepoi, Stevie Swindall, Colin Gregor, Andrew Henderson, Ruaridh Jackson
Ospreys: Gareth Owen; Johnny Vaughton, David Bishop, Andrew Bishop, Nikki Walker; Daniel Biggar, Jamie Nutbrown; Paul James, Ed Shervington, Cai Griffiths, Ian Gough, Alun Wyn Jones, Tom Smith, Marty Holah, Filo Tiatia
Replacements: Huw Bennett, Duncan Jones, Andy Lloyd, Jonathan Thomas, Rhys Webb, James Hook, Kristian Phillips