Goals in either half from Don Cowie and Martin Canning set up only their second home win of the season, and despite a late goal from Juan Escalas the Dingwall side held out for the points.
The Jags were awarded a free-kick on the edge of the box in the ninth minute but Derek Fleming shot against the wall.
County made the breakthrough a minute later when Gary McSwegan played a square ball to Cowie who took a couple of strides and picked his spot behind Kenny Arthur.
County were worth their lead, although the Jags caused a panic in the County defence when Jim Laughlan had to intercept a dangerous cross from Kenny Milne.
Then Adam Madaschi came forward to lend his forwards a hand but his long-range piledriver just cleared the upright.
County should have been 2-0 up in the 31st minute when good work from Jamie McCunnie set up John Rankin for a shot, but the former Manchester United player fired his shot against the post.
Partick's last chance before half-time came when an Andy Wilkinson cross flashed across goal but was missed by the inrushing Jamie Mitchell.
County made a bright start to the second half and McSwegan should have done better when clear through on Arthur, but the striker fired straight at the keeper.
McSwegan had another excellent chance to get on the mark, but Arthur made a blocking save with his feet to deny him.
Escalas almost levelled when his long-range effort rebounded off the inside of the post, but the next goal came at the other end in the 52nd minute following a set-piece.
Rankin flighted the ball over and Canning rose to head high into the net from six yards.
County were in full flow now and Alexander Burke forced a finger tip save from Arthur, before the Jags keeper was at full stretch to tip away a fierce drive from John Robertson.
But the Jags hit back and Escalas pulled one back in the 77th minute when a poor headed clearance from Robertson let him in to fire into the net from close range.
The visitors went all out for an equaliser and it took some desperate defending to keep them out.
County broke in injury-time and substitute Steve McGarry should have wrapped it up after a lung bursting run from Rankin, but he shot over with only Arthur to beat.