The on-loan Cardiff midfielder put the Staggies in front at Firhill against the run of play after caressing a 23rd-minute opener past Scott Fox in the Thistle goal.
But the Maryhill men responded in style as a double from new striker Lyle Taylor and Stevie Lawless' close-range effort early in the second half looked to have put them on course for their first league victory in Glasgow since beating Morton last April.
However, County left-back Ben Gordon immediately pulled one back before Kiss capitalised on Jags' jitters to slide home the equaliser with a quarter of an hour left.
County had to play the last six minutes with 10 men, though, after substitute Stuart Kettlewell was shown a straight red less then a minute after coming on.
The draw means Derek Adam's 11th-placed Highlanders remain just a point behind Partick as they both battle to avoid an end-of-season play-off survival fight.
A determined start by Thistle required some quick reactions from Staggies goalkeeper Michael Fraser as he palmed away Kallum Higginbotham's volley after the Firhill forward almost benefited from hesitant County defending within the first five minutes.
That was a great save but Conrad Balatoni's headed miss from Lawless's cross after Aaron Taylor-Sinclair had done well to set him free on the left was simply careless.
But just as the home fans began to taunt the visitors for their lack of ambition, County stuck it to them by conjuring a goal of real quality.
Graham Carey sent a pass down the left flank for striker Jordan Slew, who knocked it inside to Tidser. The former Morton man then shunted the ball along again for Richie Brittain on the edge of the Thistle box and the skipper's lay off was perfect for Kiss to plant a smacker into the bottom corner of Fox's net.
But Thistle did not let that setback deter them and a flowing counter saw them level on 28 minutes.
Evangelos Oikonomou's pass was intercepted by Jags right-back Stephen O'Donnell and his pin-point pass sent Lawless free through the middle. It was another precision through ball from the playmaker that then placed Taylor into shooting territory and he beat Fraser with the help off a nick off Scott Boyd's heel.
And after Fox did well to deal with Brittain's sneaky near-post attempt from a free-kick, Thistle struck again on 38 minutes.
Aaron Muirhead and Taylor-Sinclair combined wide on the left, with the left-back sweeping a first-time cross into the six-yard box just in time to meet Taylor's clever run. He was surrounded by Boyd and Brian McLean but still found the space to send his header past Fraser.
And it took Partick just 90 seconds into the second half to notch their third after Higginbotham and Elliott remained calm in the box as they exchanged passes before setting up Lawless to prod home at the near post.
But County responded immediately with another goal themselves. Taylor-Sinclair failed to get enough distance on his goal-line clearance after Brittain beat Fox after climbing well to meet Carey's deep cross with a header and the loose ball was smashed home by supporting defender Gordon.
The visitors survived a scare on the hour as Taylor-Sinclair's cross drifted on to Fraser's crossbar while Brittain came within an inch of netting a free-kick at the other end.
With the home side just about clinging on to that long-awaited home win, County sensed nerves in the Jags ranks.
And the pressure told on 76 minutes as Kiss latched on to Brittain's through ball before calmly beating Fox for the second time.
County had to play out the final few minutes a man down as Kettlewell - just on for Gary Glen - slid in dangerously on Ross Forbes to earn himself an instant dismissal.
O'Donnell came close with the last kick of the game but the visitors stood strong to claim a point.