The Borderers produced stout defence to deny the champions a bonus point and their own quest to salvage a consolation point was foiled when Mike Adamson slid in for a try in the ninth minute of stoppage time.
Hawks' other touchdowns were claimed by Steve Duffy and skipper Mark Sitch, while Adamson added all three conversions and a penalty double.
Karne Kaufana snapped up the only Melrose counter, with Scott Ruthven keeping them in the hunt by slotting a conversion and a penalty, James King landing their other kick.
Biggar halted their slide by outgunning inconsistent Ayr 34-10 in tricky conditions at Hartreemill - with the boot of Quintin Sanft proving crucial.
The playmaker put over two conversions, three penalties and a drop-goal double to kill off the challenge of the seasiders.
There were also tries for Richard Wood, Lee Kibble and Gary Walkingshaw, while scrum-half Alasdhair McFarlane sneaked in for Ayr's only touchdown. James Noonan converted and added a penalty.
Aberdeen GSFP produced a stirring second-half rally to salvage a four-try consolation bonus point in an all-action tussle at Myreside.
At 24-3 down they looked dead and buried, but they refused to buckle completely against the powerful home unit who eventually ran out 38-25 winners. To compound the visitors' frustrations, they lost flanker Grant Strang with a suspected broken arm.
The Sonians' tries were snapped up by Carl McWilliam (2), Steve Lawrie, David Payne, Murray Bringhurst and James Easton, with Will Campbell adding four conversions.
The Aberdeen scores were claimed by Colin Philips, Mark Ozich, Murray Low and Rod Seib, Keith Oddie slotting a conversion and penalty.
Hawick edged out relegation-haunted Stirling County 9-3 in a dour encounter at Mansfield Park.
The Greens dominated the exchanges in the first half, but could not cash in on their constant pressure. They had to rely on three penalties by Craig Neish for their success, with Brian Archilbald landing a single strike for County.
There was only one clear-cut try opportunity during the 80-minute slog, but when Neil Douglas spilled the ball across the line it just about summed up the match.
BT Premiership One
Currie 21 - 28 Boroughmuir
Glasgow Hawks 27 - 13 Melrose
Hawick 9 - 3 Stirling County
Heriot's 19 - 5 Stewart's Melville FP
Watsonians 38 - 25 Aberdeen GSFP