But then came a blistering three minutes in which County conjured up three touchdowns courtesy of Ali Mackie, Andrew Goudie and Grant Hughes, all converted by Goudie.
The shellshocked Glasgow men rallied for Thompson to chip over a second penalty to reduce the deficit at the break and they clawed their way to within one point when Jack Macready crossed the line, Thompson again adding the extras.
Hawks then forged ahead when James Couper scampered over, Thompson maintaining his excellent kicking form to convert before adding a penalty.
Goudie set up a try for Ross Bundy with three minutes remaining which brought Stirling in turn, to within one point, though Goudie sent the conversion wide.
County then bagged the bragging rights with a Matt Emmison try in the 84th minute.