In a compelling 90 minutes of football, the visitors went ahead through Graham Carey following a slack pass back by Callum Tapping.
Callum Paterson levelled before the break but the impressive Melvin De Leeuw restored County's lead with a fine finish.
The Staggies were reduced to 10 men when captain Richard Brittain was shown a second yellow card and Stevenson levelled with two minutes left.
In a lively start, De Leeuw had a shot from distance that was comfortably held by goalkeeper Jamie MacDonald.
Hearts were playing their part too and Jamie Walker picked out Scott Robinson inside the box with a piercing diagonal ball. The midfielder managed to get goal-side of left-back Branislav Micic but his chest control allowed goalkeeper Mark Brown to come out and smother the ball.
The hosts were looking to Walker for creativity and he fired a low left-footed effort just wide in the 17th minute.
The left-winger was involved in Hearts' next attack. Accepting a pass from Robinson, he cut inside before unleashing a deflected shot that looped just wide of the post.
Hearts then shot themselves in the foot in the 23rd minute with a huge lapse in concentration that allowed Carey to finish.
Tapping tried to pass back to his goalkeeper but Carey pounced and coolly slotted the ball beyond MacDonald from inside the box.
Hearts came close to levelling just seconds later but Robinson's downward header from a Jamie Hamill cross was pushed behind by the outstretched Brown.
The home side levelled in the 28th minute through Paterson. Robinson's corner was flicked on by Danny Wilson and Paterson, who had his back to goal, managed to swivel and hook a brilliant left-footed half volley into the corner of the net.
County came back again and De Leeuw cracked a 20-yard shot off the underside of the bar in the 34th minute.
And just seconds later, another slack pass back by Hamill was collected by Kevin Luckassen but his left-footed shot was blocked by MacDonald.
A County counter attack a minute before the break produced another great chance but MacDonald pushed Rocco Quinn's effort from inside the box over after he was teed up by Gary Glen.
But County regained their lead five minutes after the restart. Glen cut the ball back to De Leeuw from the goal-line and the Dutchman turned before drilling a low-footed shot into the corner of the net from 15 yards.
Hearts went about trying to level for a second time and Jordan McGhee headed just over from a Kevin McHattie free-kick.
County were reduced to 10 men in the 62nd minute after Brittain picked up his second yellow card for a late tackle on Walker at the edge of the area.
Hamill took the free-kick and his low effort around the wall was pushed behind by Brown.
Hearts pressed again and substitute David Smith shot over from close range before keeper Brown held on to Stevenson's effort from 20-yards.
The Gorgie outfit made their numerical advantage count two minutes from time when Stevenson bundled the ball home from a corner.
Smith's delivery was knocked down by Wilson and the forward forced the ball home inside a congested six-yard box.
The drama was not over, however, as Paterson crashed a header off the bar from a Smith free kick.