Neil Lennon's men must now stage another comeback and overcome another two-goal deficit to rescue their Champions League play-off when Shakhter Karagandy come calling on Wednesday.
Celtic made no fewer than six changes from the side who suffered the shock loss to the Kazakhstan champions in midweek for the visit of table-topping Inverness.
Beram Kayal, Matthews, Efe Ambrose, Tony Watt, Amido Balde and Tom Rogic were all added to the side.
Joe Ledley, Steven Mouyokolo and Mikael Lustig were named on the bench and James Forrest, Kris Commons and Georgios Samaras, who were all injury doubts beforehand, were absent from the squad.
In contrast, Inverness named the same 11 players who defeated Motherwell last week for the long trip to Glasgow.
The visitors had an early chance to pile on the pressure when Billy McKay was fouled 25 yards out and Doran's resultant free-kick swerved just past the post.
At the other end, Matthews picked out Balde for a bullet header that was straight at Dean Brill and confidently claimed by the Caley Thistle goalkeeper.
David Raven then failed to clear a long punt from Kayal that fell kindly for Watt, but Brill came to the rescue again to block the point-blank shot.
Celtic Park did not have to wait too long for the opener but it was the band of travelling supporters who were celebrating with 14 minutes on the clock.
Fraser Forster was left grasping at fresh air when Doran unleashed a superb curling effort from 20 yards that fell beyond the reach of the Hoops keeper and in off the far post.
Celtic tried to respond and Watt forced another decent save from Brill with a drilled shot at the near post that the keeper turned past the upright.
The Hoops' injury woes intensified when Kayal was forced out of the action and replaced by Ledley after 23 minutes.
Celtic came close to levelling when Ambrose met Mulgrew's free-kick with a powerful header that skimmed the crossbar.
But the home side found themselves trailing further after 35 minutes when Foran nodded home McKay's mis-hit overhead kick that appeared to be going wide.
Celtic managed to pull a goal back four minutes before the break when Mulgrew poked over the line from an Emilio Izaguirre cut-back.
Clearly boosted by the goal, the Hoops dominated the opening spell of the second half and were not too far away from levelling when Balde sent a header fizzing inches over the crossbar after meeting a teasing ball from Watt.
Inverness posed less of a threat after the break and Doran made a half-hearted claim for a penalty after going to ground in the box under pressure from Virgil van Dijk but there was no spot-kick from referee Craig Thomson.
Celtic still looked more likely to find the back of the net again and Balde drove into the box before seeing his low effort smothered by Brill at the second attempt.
The equaliser came with 82 minutes on the clock when a low drilled shot from Matthews beat the Caley Thistle goalkeeper, sparking wild celebrations along the touchline from boss Lennon.