Craig Curran got County off to a flyer with a goal in the second minute, but Mark Reynolds equalised mid-way through the first half with a fine finish from just inside the 18-yard box.
The Dons dominated for long spells in the second half and Shay Logan headed home with just under 20 minutes to spare to win the game 2-1.
County took the lead after just 82 seconds when Curran glanced home a Chris Routis free-kick with a downward header.
Curran had a half chance in the sixth minute when he headed over after some head tennis in the box.
It was all County in the opening spell and Routis flashed a shot over the crossbar following a Jason Naismith cross.
Aberdeen’s first attempt at goal came in the 13th minute when Kenny McLean disposed Jamie Lindsay 25 yards out but his shot was well saved by Scott Fox.
The visitors were coming more into the game and they equalised in the 23rd minute when a Greg Tansey corner was only partially cleared and when the ball fell for Reynolds he lashed the ball into the net from 16 yards.
County had a chance to restore their lead in the 34th minute. Michael Gardyne did well to get a cross over from the left which fell for Naismith but he failed to find the target with a weak effort.
Aberdeen won a free-kick in a dangerous position in the 51st minute but Considine’s ball into the box was missed by everyone.
The visitors came close again three minutes later when Fox pulled off a fantastic one-handed save from Greg Stewart.
The Dons came close again in the 58th minute when a Tansey free-kick was parried by Fox but May could only scoop the rebound over from close range.
The pressure paid off in the 71st minute when Gary MacKay-Steven fired the ball over from the byline and Logan rose at the back post to nod home.
County were only seen in flashes but Jim O’Brien forced a fine save from Lewis with a low drive from 25 yards, which the keeper turned for a corner.
Nicky Maynard, who replaced May late on, came close with his first touch following a corner when he headed over from close range.
County came desperately close to a leveller in stoppage time when O’Brien cracked a shot from 25 yards against the crossbar.
But Aberdeen held on for the points.