Twelve months after losing at the same stage in last season's competition to the same opponents, the Dingwall side reached the next round on this occasion, but they were once again made to fight all the way by plucky Stranraer.
The Highland side fell behind after just three minutes when the ball broke for Stephen Stirling inside the penalty area, with the midfielder sliding a quick pass to Sean Winter, who sent a looping effort into the top-right corner of the net.
County equalised after 15 minutes after goalkeeper David Mitchell spilled Jordi Balk's cross and Melvin De Leeuw rolled the loose ball into the net.
The Dingwall side then pressed hard for the winner and it came in the 69th minute. In a sweeping move County switched the ball wide to Joe Cardle whose stinging drive was pushed away by Mitchell, with the goalkeeper also denying De Leeuw from the rebound. However, Liam Boyce followed up to score.