Neil Lennon's men will be crowned champions if they beat Partick on Wednesday night after Aberdeen came unstuck at Victoria Park.
Melvin De Leeuw headed home Graham Carey's corner in the 35th minute to give Ross County the lead.
And although Adam Rooney pulled one back with 13 minutes remaining, the visitors failed to complete the comeback to keep the pressure on Celtic.
County thought they had taken the lead in the 21st minute when De Leeuw flicked the ball in for Yoann Arquin, who shrugged off a challenge from Alan Tate and coolly tucked the ball past Jamie Langfield. However, the goal was ruled out for offside.
The home side came close again in the 32nd minute when a De Leeuw effort was blocked on the line by Logan.
But County did take the lead three minutes later when the home side were awarded a free-kick after Andy Considine fouled Erik Cikos. The Dons' defender was booked and when Carey curled in his delivery De Leeuw rose to glance a header into the net.
County were rampant and Arquin was sent clear on goal by De Leeuw - but Langfield raced from his line to block his shot.
The visitors looked stronger and after Barry Robson crashed a 30-yard drive off the crossbar a Robson shot at the near post hit Yann Songo'o before going behind for a corner. Robson claimed the County defender had handled but referee Brian Colvin was having none of it.
A Cammy Smith header clipped the post and Shaughnessy fired the rebound straight at Mark Brown, before County went on the counter-attack and Carey looked to have been impeded by Considine inside the box - but play was waved on.
It was County's turn to apply the pressure and a poor clearance from Langfield fell to De Leeuw, whose long-range shot was gathered at the second attempt before Arquin could get a touch.
The next goal came at the other end. After a flurry of activity around the County goal which included shots from Jack and Considine, which were saved, Rooney drilled the ball into the corner of the net for his eighth goal in 11 games.
With time running out Carey was through on goal and about to shoot when he was taken down by Considine on the edge of the 18-yard box.
The Aberdeen player was shown a straight red card and Brittain sent his free-kick into the wall and behind.
Deep in injury time Songo'o got on the end of a Brittain free-kick but Langfield made a point-blank save to deny him the winner.