The Jambos, who were docked 15 points after entering administration last summer, wiped out the deficit after Scott Robinson grabbed a late winner at Dingwall.
The midfielder kept his composure in the box before chesting the ball down and volleying beyond helpless keeper Michael Fraser with just 10 minutes remaining.
The victory was well deserved for Gary Locke's side who looked like paying for a number of missed opportunities before Robinson sent the visiting contingent wild.
Sam Nicholson, Callum Paterson, Dale Carrick and Kevin McHattie all squandered fine chances to open the scoring for the league's bottom side before Paterson settled their nerves when stabbing the ball beyond Fraser six minutes before half-time.
But in a rare County attack, on-loan Blackburn defender Yann Songo'o headed the home side level after the Hearts rearguard failed to deal with a long ball into the penalty area.
County came more into the game after the break but it was Hearts who almost regained the lead when Carrick shot straight at Fraser when through one-on-one with the home keeper.
Visiting keeper Jamie MacDonald kept the scoreline level when pulling off a stunning save to deny Songo'o his second of the match with 54 minutes on the clock.
And as both sides looked to be settling for a point apiece, Robinson grabbed his first goal of the season to lift the Jambos onto two points for the campaign, 19 points adrift of second-bottom Partick Thistle.