Stephen O’Donnell grabbed the winner in the 52nd minute to send Alan Archibald’s side within three points of securing their Premiership status for another season.
County, who had collected 25 points from their previous nine matches, took the lead when midfielder Michael Gardyne netted from a tight angle in the 21st minute but Frederic Frans levelled with a 20-yard shot just five minutes later.
Partick dominated for most of the opening period but Ryan Stevenson missed a number of chances to add to their solitary strike.
Stuart Bannigan and Callum Booth went close as Thistle continued to look the better side and, after Lyle Taylor fired narrowly wide with their latest squandered chance, O’Donnell’s rising strike put them 2-1 ahead.
County were unable to respond during the final half hour with O’Donnell coming close to extending Partick’s advantage in the 70th minute when his 18-yard effort was saved by home keeper Mark Brown.
With four matches left, County are six points above Motherwell, who remain favourites to occupy the relegation play-off spot, with St Mirren – 4-1 conquerors of Kilmarnock – a further seven points adrift.