Steven Lawless hit a double early in the second half after Kris Doolan had opened the scoring.
Graham Carey grabbed a consolation goal for County with a deflected shot but there was no way back.
After a cagey opening spell, the visitors created the first real threat when Lawless chipped the ball into the box for Doolan but County full-back Mihael Kovacevic did well to get in a block tackle.
But the opener was not long in coming. Thistle forced the first corner of the match in the 18th minute and Gary Fraser’s delivery was knocked on by Conrad Balatoni for Doolan to scoop the ball into the net at the back post.
County pressed for a leveller and Carey caused some concern with a dangerous free-kick into the box but the ball deflected behind for a goal kick.
Lawless was proving a handful for the County defence and he went on a jinking run in the 29th minute but his final shot was blocked on the edge of the 18-yard box.
The home side came close to an equaliser in the 42nd minute when Kovacevic found Mustafi 12 yards out but his effort was blocked in front of goal by Thistle defender Aaron Muirhead. It was the best County could offer as Thistle finished the half deservedly in front.
County made a double substitution at the start of the second half with Melvin De Leeuw and Steven Ross coming on for Branislav Micic and Darren Maatsen.
But it was Thistle who almost doubled their lead in the 47th minute when Sean Welsh’s cross into the box was headed on by Muirhead, but Ben Gordon was on the line to head clear.
Thistle almost punished a poor clearance from home keeper Mark Brown when Steven Lawless was picked out by James Craigen but the County keeper redeemed himself with a superb save.
But Brown could not keep Thistle at bay for much longer. The visitors took advantage of some sloppy play in the 53rd minute and after Stuart Bannigan was denied, Lawless was on hand to drill home the rebound.
It got worse for County six minutes later when Bannigan sent a diagonal ball across goal for Lawless to force home at the back post.
County kept plugging away and they pulled one back in the 71st minute when Carey scored with a deflected shot from outside the area.
Thistle new boy Kallum Higginbotham had the ball in the net in the final minute but was ruled offside.