Serious injury to Scotland hooker Ross Ford took the shine off Border Reivers' victory, which made them firm favourites for a Heineken Cup place.
Glasgow Warriors (0) 8:
Border Reivers (9) 14:
Tries: Danielli, Miller
Cons: MacRae (2)
Pens: MacRae (2)
Ford was stretchered off after just two minutes before Glasgow fell behind to a Calum MacRae penalty.
Simon Danielli and Andy Miller grabbed tries, while MacRae contributed another penalty and two conversions.
Glasgow's Colin Gregor missed kicks to reduce the deficit before breaking his duck and Sam Pinder added a try.
Victory increased the Borders' lead over Glasgow in their quest to become Scotland's second Heineken Cup entrants behind Edinburgh.
Yet the home side had carved out the first real scoring chance after centre Graeme Morrison intercepted an intended chip over the top by Danielli.
But Miller back-tracked to halt the move with a brilliant tackle.
And MacRae immediately booted the Borders in front with a confidently-struck penalty after Rory Lamont was guilty of playing the ball illegally on the ground.
Gregor squandered an opportunity to square the match and the visitors capitalised by opening up a 13-point advantage at the interval.
Danielli slid over in the corner for a tidy try in the wake of a scrummage on the 22m-line.
Then MacRae banged over the tricky conversion and added a penalty after Gregor had wasted another chance for Glasgow.
There was even more Glasgow frustration on the stroke of half-time when skipper Jon Petrie was bundled out of play just a metre from the target after a powerful surge featuring Pinder and Steve Swindall.
Glasgow suffered a further setback shortly after the re-start as winger Mike Roberts hobbled off with a leg problem.
The Warriors had the better of the exchanges for the next 10-minute spell but could not find a way through the watertight Reivers defence.
And their woes worsened when Miller galloped over for the second Borders try after latching on to a pin-point pass from fit-again stand-off Gregor Townsend.
MacRae slotted the extra points from wide out.
Glasgow broke their duck when Gregor rediscovered his touch.
And they gave themselves a glimmer of hope when Pinder darted over following a quick tapped penalty under the crossbar by prop Kevin Tkachuk.
But Gregor's jitters returned as he sent the conversion wide.
Glasgow: Staniforth, Lamont, Morrison, Davey, Roberts, Gregor, Pinder, Tckachuk, Lawson, Murray, Turner, Hamilton, Swindall, Petrie, Hayter.
Replacements: Howarth for Staniforth (75), O'Hare for Roberts (43), Thomson for Lawson (64), Harrison for Murray (71), Beattie for Hayter (51). Not Used: Eddie, McFarlane.
Borders: Moffat, Walker, MacDougall, Danielli, MacRae, Townsend, Cusiter, Thomson, Ford, Cross, MacLeod, Palepoi, Brown, Miller, Sititi.
Replacements: Hore for Townsend (59), Scott for Ford (3), McGee for Cross (70), Gray for Miller (54), Dalziel for Sititi (70). Not Used: McKearchar, King.
Referee: Rob Dickson (Scotland)