Adam Matthews' late equaliser prevented leaders Inverness Caledonian Thistle, so often a thorn in Celtic's side, recording another victory in Glasgow.
Caley Thistle, the last Scottish team to win at the home of the Scottish champions in November, went ahead through Aaron Doran's 25-yard drive.
Richie Foran forced in a second after 35 minutes, but Charlie Mulgrew poked in a reply before the break.
Matthews fired home, but Celtic's run of nine straight home wins ended.
Mulgrew celebrates his goal with Emilio Izaguirre against Inverness
It means Caley Thistle remain three points clear of Celtic, who had won their last 13 top-flight outings on their own patch since that defeat by Inverness, at the top of the Scottish Premiership.
A much-changed Celtic side had taken to the field looking to recover from the 2-0 Champions League play-off defeat by Shakhter Karagandy - the Glasgow side's first in seven games this season.
The absence of James Forrest, Kris Commons and Georgios Samaras was the result of injuries, while Joe Ledley, Steven Mouyokolo and Mikael Lustig also dropped out, possibly with a view to being rested ahead of next week's second leg.
One of the incoming half-dozen, Tony Watt, should have opened the scoring when he found himself one-on-one against Dean Brill, but the Scotland Under-21 striker's attempt to steer the ball into the corner was too close to the goalkeeper.
The home side were quickly made to rue that missed chance when Caley Thistle took the lead.
Doran picked up the ball 25 yards from goal and curled it beautifully past Celtic goalkeeper Fraser Forster.
Nine minutes later and Celtic manager Neil Lennon was handed another injury headache, with midfielder Beram Kayal being forced out of the action and Ledley's rest period on the bench was cut short.
Brill did well to turn a low Watt drive wide, while Efe Ambrose, who had been returned to the Celtic defence, powered a header just over the Inverness crossbar as Celtic started to ramp up the pressure in search of an equaliser.
However, it was the visitors who stretched their lead when Foran reacted first to Billy McKay's mis-hit overhead kick to force home Caley Thistle's second.
Just when it looked like another surprise victory for Caley Thistle at Celtic Park was on the cards, Mulgrew pounced on a cut-back from Emilio Izaguirre to toe-poke past Brill.
Celtic failed to produce an equaliser before the half-time whistle relieved the rising pressure on the Caley Thistle defence.
Amido Balde, handed his first competitive start for Celtic, flicked a header just over the crossbar from a Watt cross as the summer signing came close to his first goal since arriving from Vitória Guimarães.
Inverness were now rarely seen in attacking mode but had claims for a penalty waved away as Doran, who had delayed his pass to an unmarked Nick Ross, was challenged by Virgil Van Dijk.
Celtic eventually got the equaliser their probing merited when a fine through ball from captain Scott Brown gave Matthews, making a rare start following injury problems, the chance to fire past Brill.
Substitute Mikael Lustig sent a late header just wide and Inverness held on for a point.
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