Kallum Higginbotham scored the opener for the visitors after nine minutes but Ross were back on level terms shortly afterwards through Richie Brittain.
Lyle Taylor went close before the opening goal was scored when he headed a Higginbotham free-kick just wide of the target.
Taylor was also involved as the away side took the lead after he was fouled by Scott Boyd and Higginbotham stepped up to fire a 22-yard free-kick under the wall after it had jumped to try and keep the ball out.
The lead did not last for long though, with the home side back on level terms just five minutes later.
Ross earned a free-kick of their own just outside the Thistle area and Brittain smashed the ball home.
Partick continued to press as they looked to regain the lead but Prince Buaben could only shoot over after some good work in the build-up from Chris Erskine.
Taylor then had a shot deflected over before County went close at the other end, with Filip Kiss denied by Lee Mair's crucial last-ditch tackle as the Slovakian bore down on Partick goalkeeper Paul Gallacher.
Mair was involved at the other end after the restart when he blasted over at the back post and Erskine also spurned the opportunity to score when he shot just past the left post of County keeper Mark Brown.
Brown's opposite number Gallacher was also kept busy as he was forced to save well to deny half-time substitute Melvin De Leeuw.
Evangelos Oikonomou, Kiss and De Leeuw all went close to securing a late winner for the hosts but Partick held on to secure a share of the spoils.