Arbroath striker Steven Doris knocked in a late strike to earn a famous Scottish Cup draw for the Second Division side at Celtic Park.
Despite their dominance, Celtic needed a bizarre own goal from Alex Keddie nine minutes before half-time.
The Arbroath defender was left helpless as Stuart Malcolm smashed an attempted clearance against him.
Miku hit the woodwork as Celtic pressed after the interval, but Doris levelled with his deflected 87th-minute strike.
Celtic spurned chance after chance to put the game beyond the visitors but some inspired goalkeeping from Scott Morrison kept Arbroath in the game.
A wonderful late save from substitute Lassad kept Paul Sheerin's side in the tournament but the home team will rue the good chances missed by Tony Watt, Miku and Rabiu Ibrahim, all of whom should have scored at least once.
Celtic manager Neil Lennon shuffled his pack, making nine changes, with one eye on Wednesday's Champions League clash with Spartak Moscow.
And, in front of a sparse crowd, his side began searching for an opener with Miku heading over after only three minutes.
Watt saw an effort saved by Morrison, with Victor Wanyama and Scott Brown both going close.
For all Celtic's possession, they rarely troubled Morrison in the first half, although he was beaten before the break by the oddest of own goals.
A Celtic attack had petered out and the visiting captain Malcolm looked to have things under control on his own byeline. However, his attempted clearance wasn't clever, as he needlessly blasted the ball off Keddie and into the net past an incredulous Morrison in goal.
Arbroath had their first effort on goal shortly after, when Lukasz Zaluska saved Mark Baxter's 20-yard drive.
After the break, Celtic kept piling on the pressure but the Arbroath defence held firm, with Morrison in particularly fine form.
Had Ibrahim not curled a shot just over, or Miku found the net rather than the crossbar with a close range effort, things could have been different.
But with just one goal in it, Arbroath sensed the possibility of a huge shock and the goal they craved arrived in the closing minutes.
A free kick just outside the box was well hit by Doris but it was the huge deflection off the Celtic wall that took it spinning past Zaluska.
Arbroath survived a late Celtic onslaught, including three minutes of stoppage time, to book a money-spinning replay on 12 December, which may attract live television coverage.
That will certainly be the case for Celtic's next home match.
They could reach the last 16 of the Champions League against Spartak. But failure to reach the same stage of the Scottish Cup at the first time of asking against a team two divisions below them is far from the ideal preparation.