The 22-year-old struck twice in seven minutes in the early stages of the second half after Kris Doolan had given Partick a first-half lead.
Graham Carey pulled one back for Ross County but Partick held firm to move second in the table.
Partick, who began their campaign with a goalless draw with Dundee United last weekend, forced the early pressure with Stuart Bannigan having the first effort on goal.
They soon took the lead with 26-year-old Doolan diverted the ball into the net at the back post after Gary Fraser's delivery was knocked on by Conrad
County tried to respond but efforts from Darren Maatsen, Graham Carey and Grant Munro all failed to hit the target.
Rocco Quinn also failed to work Partick goalkeeper Scott Fox when he fired wide of the goal midway through the first half.
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 Mihael 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.
Thistle 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 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 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.