The Gers dominated the match for large periods, but were wasteful in front of goal, especially in the second period.
Ross County were broken down in the 35th minute as John Lundstram found the bottom-right corner of the net from the edge of the box.
Fashion Sakala, never from the action for Rangers, set up the midfielder for the goal.
Elsewhere in the Premiership, Kilmarnock fought back against Motherwell to overturn a 1-0 score-line into a 2-2 draw.
Kevin van Veen struck in the first period for the home side, before Callum Slattery made it two in the second.
The Killie benefited from a heavily deflected Daniel Armstrong shot off Paul McGinn in the 73rd minute to get back in the game.
The comeback was completed two minutes later through Liam Polworth, who bagged an absolute belter.
These results mean Kilmarnock leapfrog Ross County into 10th place, while Celtic now need to react to Rangers picking up another three points.
