Leigh Griffiths fired Celtic into a 23rd-minute lead with a sizzling shot that briefly lit up a subdued crowd at Celtic Park.
However, a sensational strike from Stewart Murdoch levelled the match in the 64th minute as Celtic failed to take full advantage of Aberdeen's 3-0 defeat to St Johnstone on Friday.
The full-time whistle was greeted with jeers from the home support, who vented their angry at departing manager Ronny Deila and his players.
Despite this, the Hoops are now nine points clear with four games remaining and could clinch the championship with a win away to Hearts next Saturday if Aberdeen fail to pick up maximum points in their match with Motherwell.
Deila's announcement that he will depart the Glasgow giants in the summer had done little to boost the numbers or atmosphere within Celtic Park following last weekend's Scottish Cup semi-final defeat to Old Firm rivals Rangers.
However, the home crowd finally came to life midway through the first half when Griffiths rifled the Hoops in front.
The Scotland striker was allowed too much room on the edge of the box after collecting a pass from Colin Kazim-Richards and he drilled an unstoppable strike into the top corner.
Minutes later, Australian star Tom Rogic somehow missed a chance to double Celtic's advantage as he fired wide from five yards out after getting on the end of a cross from Patrick Roberts.
County 'keeper Scott Fox then got down low to tip a fierce effort from Nir Bitton round the post while at the other end Craig Gordon held on to a low drive from Ian McShane.
County's Liam Boyce then missed a chance to haul his side level as the unmarked striker sent a header straight to Gordon from eight yards out.
The visitors passed up another opportunity five minutes after the break when Michael Gardyne blasted over from six yards out under no pressure after collecting a headed knock-on from Alex Schalk.
A 25-yard screamer from Murdoch levelled matters in the 64th minute after a period of sustained pressure from County.
An attempted clearance only went as far as Murdoch who took a touch before smashing a shot beyond Gordon, although the goalkeeper complained about a County player in an offside position blocking his view.
The Celtic fans thought Kazim-Richards had put the Hoops in front again but the Turkish striker fired wide and was quickly replaced by Kris Commons.
Dutch striker Schalk then sent a powerful strike from 25 yards inches wide as County came close to snatching a winner of their own.
Elsewhere, Dundee United edged closer to relegation from the top flight as Hamilton Academical sealed a 3-1 win over Mixu Paatelainen's side to leave them eight points adrift at the bottom of the table.