They had brushed off a series of setbacks to narrow the gap to a single point with less than 15 minutes left - only for Hawks to inflict heartbreak with a stoppage-time try.
The home fans sensed an upset in the opening stages when Barry Edgar confidently banged over three penalties in quick succession, but their celebrations quickly turned sour as they squandered two clear-cut try chances.
First, Alan Bulloch's feed to Andy Rushforth was marginally forward - then David Jamieson committed the same offence when Rory Watson had a clear path to the line.
Having weathered the early storm, Hawks burst back to grab control with a brace of touchdowns when GHA were reduced to 14 men through the sin-binning Ian Smith.
Matt Smith was driven over for the first following a sustained spell of pressure, Murray Strang adding the conversion.
The second try was almost a carbon copy as scrum-half Kenny Sinclair claimed the vital touch following a well-controlled line-out and pack surge.
Even worse was to come for GHA just before the break. This time, Hawks mauled down the middle of the pitch before being blocked close to the target.
The ball was flashed out from Mike Adamson, giving Wes Henry plenty of time and space to cruise over and open up an eight-point gap.
Niggles began to creep into the contest and a likely scoring move by Hawks was halted by referee Mike Hall when several players were involved in a mass brawl.
The visitors kept up the momentum and stretched their advantage thanks to a touchdown by Steve Begley - but GHA refused to buckle and they earned a glimmer of hope when Rushforth ploughed through in the wake of a 50-metre dash by scrum-half Ally Macfarlane.
And the encounter was dramatically back in the melting pot in the 66th minute as prop Gavin Walsh burrowed over for another crucial score, with Edgar adding the tricky conversion.
But Henry's last-gasp effort, goaled by Strang, ended their plucky resistance.
You can see photos from the game here