Kallum Higginbotham put the visitors ahead from a free-kick in the ninth minute, only for Richard Brittain to level matters from another set-piece five minutes later.
With St Mirren winning at home to Kilmarnock, the result saw both teams drop a place in the table - with just rock-bottom Hearts below them.
Partick started with a spring in their step and Lyle Taylor flashed a header wide of Mark Brown's right post following a Higginbotham free-kick.
And Thistle did not have to wait too long to open the scoring, doing so from their next set-play. Scott Boyd was booked by referee Craig Thomson for a tackle on Taylor as the big forward bore in on goal, and Higginbotham drove the resulting free-kick through the wall and into the corner of the net.
But County were back on level terms five minutes later when Lee Mair fouled Filip Kiss just outside the 18-yard box, and Brittain stepped forward and curled the free-kick past Paul Gallacher.
Partick pressed to restore their lead in the 25th minute, when Chris Erskine beat Boyd on the left but his cut-back was swept over the crossbar by Prince Bauben.
At the other end, Michael Tidser opened up the Thistle defence in the 34th minute with a pass for Evangelos Oikonomou but the Greek full-back delayed his cut-back and the ball was knocked behind for a corner.
County had another chance three minutes later when Tidser swung over a cross from the left, but Yoann Arquin's header flew well over from six yards.
The home side continued to probe, and Kiss was almost in on goal a couple of minutes from the break but he failed to control as through ball from Jordan Slew at the vital moment.
County made a change at the start of the second half, with Melvin De Leeuw coming on for Slew.
But it was Partick who were the team on the ascendancy early in the second period and, after Mair blazed a shot well over at the back post, Erskine latched onto a Higginbotham pass and drove a foot wide of Brown's left post.
The visitors were well on top early on and Boyd had to make a saving tackle in front of goal before Erskine could pounce.
County weathered the storm and a diagonal ball from Yann Songo'o found De Leeuw, who cut inside and let fly at goal but Gallacher got down well to smother the ball at his near post.
It was County's turn to press and Oikonomou got on the end of a diagonal ball from Arquin but his shot was deflected behind for a corner.
The hosts were going all out for a winner but, after Kiss just failed to connect with an Oikonomou cross, De Leeuw's shot from inside the area was blocked by Stephen O'Donnell.