Aberdeen dominated for long spells and it took some desperate defending from County to keep them at bay.
The only goal of the match arrived in the 56th minute when Niall McGinn crossed into the box and Quinn, under pressure from Adam Rooney, sliced the ball into his own net.
County forced a corner in the first minute and when the ball came back to Graham Carey his cross into the box was back heeled away by Dons’ defender Mark Reynolds.
Michael Gardyne then went on a rampaging run at the Dons’ defence before he run out of space and support, while at the other end a near post header from Rooney was tipped away by County goalkeeper Mark Brown.
It was a pulsating contest and the visitors came close (22) when Brown failed to hold a Rooney knock down from a McGinn free kick but Quinn stepped in to clear the danger.
The visitors pressed as the half wore on and Shay Logan cut in and let fly with a vicious shot which just cleared the crossbar.
County finally woke up in the 39th minute with Gardyne swinging over a cross into the box but Quinn’s glancing header dropped wide of the far post.
The pace moved up a gear at the start of the second half and Johnny Hayes got behind the County defence to send over an inviting ball but Darren Barr was on hand to head clear.
But it was the Dons to scoreb the crucial goal (56) thanks to Scott Boyd giving the ball away cheaply and when McGinn whipped the ball into the box Quinn steered the ball into his own net while attempting to clear.
The visitors piled forward and Paul Pawlett was unlucky with a shot which rebounded off the junction of the post and bar. It was one-way traffic and Boyd did well to stop a goal-bound effort from Rooney as the Dons turned the screw.
Liam Boyce almost made an instant impact when he replaced Gardyne (67) when he ballooned his first touch high and wide, but Aberdeen remained dominant and Boyd twice threw himself in front of shots to block.
It was an all-out assault on the County goal Boyd came to the rescue again as Hayes dinked into the box.
Willo Flood had the chance to kill it off in the dying seconds but he blazed his shot over the bar.