It took the Bhoys 77 minutes to strike at Victoria Park, but they managed to nab all three points to give themselves a six-point advantage over the rest of the field for the time being.
County tried to cause the league leaders some problems early in the first half when Michael Gardyne managed to get in behind the defence.
However, the striker's pass for Davies Keillor-Dunn went just ahead of his teammate and Celtic goalkeeper Craig Gordon managed to time his run to collect the ball on the edge of the area.
Keillor-Dunn was involved again a few minutes later when he received the ball on the edge of the penalty box and attempted a shot with his first touch, but he missed the target.
At the other end, Callum McGregor unleashed a well-executed pass towards Stuart Armstrong, but the move broke down until the midfielder won the ball back and got tackled by Christopher Routis, who escaped giving away a penalty.
Brendan Rodgers's side continued on the attack, but County stood firm by piling bodies into their own half to prevent their opponents from finding a way through.
The visitors came close to finding the breakthrough early in the second half when McGregor whipped in a corner kick, only for Mikael Lustig to smash the ball into the side-netting from the near post.
Frustration grew for Celtic as Scott Brown attempted a long-range effort, which went high and wide, but their persistence paid off when Griffiths stepped up to take a free kick.
The forward lifted the ball over the top of the wall and it dropped into the right top corner, giving the Bhoys a deserved lead on away soil.
Olivier Ntcham came close to making it two when County failed to clear the danger and the midfielder smashed a volley onto the woodwork.
In the end, Celtic were able to hold on to their lead and continue their charge at the top.