Borders (7) 22
Tries: Morton, Cross 2.
Cons: Morton, Laidlaw.
Ospreys (10) 16
Pens: Henson 2.
Drop Goals: Connor.
Geoff Cross snatched two tries at Netherdale to provide the platform for a rare Borders triumph in the Celtic League.
Home coach Tony Gilbert will hope that victory over the Ospreys will provide a springboard to leap away from the bottom end of the table.
The Welshmen looked the hungrier outfit in the opening session and deservedly broke the deadlock with a try in the 14th minute.
They mounted a strong pack drive after a lineout, leaving flanker Gavin Thomas to power over, with Gavin Henson slotting the conversion.
Borders responded in positive fashion against the combined Neath/Swansea outfit.
Lock Colin Stewart embarked on the initial charge to set up a fine try by Cross in the 32nd minute.
Gareth Morton made amends for an earlier penalty miss by adding the extras, only to see Henson edging his side in front with a penalty just before the break.
Morton burst back into the spotlight to grab the second Borders try after a surge by Chris Cusiter.
Ospreys rallied with a further Henson penalty and a snap drop-goal from Shaun Connor.
But Borders reclaimed the initiative when Cross completed his double, Clark Laidlaw adding the conversion .
Laidlaw then slotted a last-gasp penalty to make sure of the win.
Borders: Morton, Moir, Stuart, Utterson, Murray, Laidlaw, Cusiter, Graham, Ford, Cross, MacLeod, Stewart, Feather, T. Walker, Kane.
Replacements: Scott, Dunlea, Weir, Rennick, Armstrong, Turnbull, Paterson.
Ospreys: Henson, Pugh, Tiueti, Gibbs, Morris, Connor, A. Williams, Jones., B. Williams, Millward, Newman, Tait, Bater, Bonner-Evans, Thomas.
Replacements: Seveali'i, Durston, R. Wells, Littlehills, Griffiths, Clatworthy, Bateman.
Referee: Simon McDowell (Ireland)