Glasgow edged to a 26-25 victory against the Celtic League's bottom club.
Glasgow (16) 26
Borders (16) 25
Pens: Morton 5
Drop Goals: Laidlaw.
But Borders' agony at the foot of the table was compounded when kicker Gareth Morton lost his nerve in the closing stages, missing three penalties.
Relieved home coach Hugh Campbell said: "I was happy to hear the final whistle. We gave away far too many penalties."
An eventful first half opened with a batch of penalties.
Borders took the lead with a successful effort from Morton in the second minute.
Glasgow responded and Dan Parks put over a long strike.
But three minutes later referee Paul Adams spotted an offence right on the halfway line and Morton struck it home.
Glasgow scored the night's opening try on 14 minutes, Andy Wilson feeding Sean Lamont, who touched down.
When Glasgow's Lamont was sin-binned, Borders' game started to gel.
Parks had a clearing kick charged down and from the resulting line-out Morton
came into the line to touch down under the bar and convert.
Glasgow pulled the score back to 16-16 by the interval thanks to two more Parks penalties.
And, as the tension built, the hosts were just about able to sneak a win.
Glasgow: Stuart Moffat, Rory Kerr, Graeme Morrison, Alan Bulloch, Sean Lamont, Daniel Parks, Kenny Sinclair, Matthew Proudfoot, Scott Lawson, Lee Harrison, Andy Hall, Nathan Ross, Roland Reid, Paul Dearlove, Andrew Wilson.
Substitutes: Simon Gunn, Euan Murray, Joe Beardshaw, Rory McKay, Mark McMillan, Calvin Howarth, Jon Steel.
The Borders: Gareth Morton, Craig Moir, Jono Stuart, Kevin Utterson, Cammie Murray, Clark Laidlaw, Chris Cusiter, Kelvin Todd, Stevie Scott, George Graham, Doddie Weir, Colin Stewart, Andy Rennick, Dan Kane, Tony Walker. Substitutes: Ross Ford, Geoff Cross, Scott MacLeod, Kelly Brown, Iain Fairley, Scott Paterson, Calum MacRae.