Stuart Bannigan opened the scoring late in the first half, with the hosts adding three more goals after the break.
Although Thistle created most of the chances early on, County enjoyed plenty of possession.
Thistle came within inches of breaking the deadlock midway through the first half when Gary Fraser struck a post from inside the box.
At the other end, Rocco Quinn headed wide from close range before Bannigan fired Thistle ahead with a rising shot from 10 yards after latching on to Kallum Higginbotham’s assist.
County’s Graham Carey hit the bar with a long-range effort on the stroke of half-time, while Ryan Stevenson missed a great chance to add a second on 51.
Yet Thistle needed just three more minutes to double their advantage when the unfortunate second-half substitute Ben Frempah put through his own net after failing to deal with Stephen O’Donnell’s cross.
With 17 minutes remaining, Fraser located the bottom right-hand corner of the net after collecting Christie Elliott’s pass to make it 3-0, with O’Donnell completing the scoring in the final minute of normal time with a low strike from 12 yards.