The Highland minnows, who gained promotion to the top flight this season, were cruelly denied a famous win over the Hoops when Kris Commons scored dramatically in injury-time following Richard Brittain's 49th minute opener to make it a 1-1 draw.
Celtic boss Neil Lennon had a long list of injuries for the Week 3 clash but that could not be used an excuse as the Staggies have a much smaller payroll and a lot less top flight experience. The draw left Celtic sitting fifth on the table with a win and a draw in their two matches whilst Ross County moved up to ninth with a third successive draw.
In other SPL matches, Heart of Midlothian were also denied victory when Inverness Caledonian Thistle netted in injury-time to force a 2-2 draw.
Arvydas Novikovas opened the scoring for Hearts in the 15th minute before John Sutton added a second from the penalty spot late in the first half. However, Caley hit back prior to the hour-mark via a Andrew Shinnie spot-kick, after Richie Foran was sent off, before Conor Pepper saved the day for the visitors late at the Tynecastle Stadium.
A Jamie Murphy double gave Motherwell a 2-1 away win over Kilmarnock who were served by a 29th minute Gary Harkins goal whilst Aberdeen produced the same scoreline at St Johnstone thanks to goals from Isaac Osbourne and Jonny Hayes with Nigel Hasselbaink on target for the Saints.
A third 2-1 away win came courtesy of Hibernian who beat St Mirren thanks to a double from Leigh Griffiths but the Buddies were unlucky as Paul McGowan missed from the penalty spot before Steven Thompson gave them hope just after the hour-mark.
And finally, Dundee United cruised past derby rivals Dundee with a 3-0 home victory thanks to an early opener from Gavin Gunning and a Johnny Russell double with the striker also being sent off, along with Dundee captain Stephen O'Donnell, for violent conduct with about 20 minutes remaining.
The Arabs jumped to the top of the table with their second 3-0 victory of the season, one point clear of Hearts and Motherwell on five points whilst St Mirren, Celtic, Aberdeen and Hibs all have four.