The substitute netted 14 minutes from time after Liam Boyce had cancelled out Greg Kiltie’s opener for Kilmarnock to ensure County ended their season on a high.
The visitors almost got off to the perfect start when Boyce had the ball in the back of the net after just three minutes only to see his effort ruled out for offside.
And the Staggies continued to dominate early possession with Kilmarnock goalkeeper Conor Brennan – starting in place of Craig Samson – denying Michael Gardyne’s right-footed effort.
Kilmarnock began to come back into the contest as the first half drifted past the midway point with Aaron Splaine and Josh Magennis both seeing shots blocked.
Samson was called into action three times in seven minutes to deny Jamie Reckford, Scott Boyd and Martin Woods but against the run of play Kilmarnock opened the scoring.
Woods conceded possession in the 34th minute, allowing Magennis to burst forward before finding teenager Kiltie who made no mistake with his shot which beat Antonio Reguero’s dive.
Kilmarnock almost doubled their advantage four minutes later but this time Reguero managed to spectacularly claw away a stunning long-range effort from Splaine.
Quinn replaced Woods at half-time but it was Boyce who created County’s first opening after the break, firing a shot at goal from distance which was turned wide by Samson.
A succession of corners followed before Boyce got County back on level terms, firing across Samson and into the top left corner from close range after receiving the ball from Marcus Fraser.
Magennis almost restored Kilmarnock’s lead, forcing Reguero into another handy save before Raffaele De Vita and Craig Curran were unable to find the target at the other end.
But Quinn ensured County signed off their season with a win after a neat finish into the bottom right corner after collecting a pass from Boyce.
Lee Ashcroft almost salvaged a point for Kilmarnock in added time but was unable to beat Reguero with his header as County claimed all three points at Rugby Park.