An injury-time strike from substitute Georgios Samaras took Celtic back 15 points clear of Motherwell at the top of the table but not before a remarkable game unfolded.
Midfielder Kris Commons fired the home side into the lead with around 12 seconds gone but the expected romp did not materialise.
Dons striker Scott Vernon levelled with a minute of the first half remaining and strike partner Josh Magennis grabbed a double after the break to set the visitors on the way to what would have been an unexpected victory.
However, with SPL defeats to Motherwell and Ross County fresh in their minds, Celtic showed the spirit of champions and Charlie Mulgrew reduced the deficit before Gary Hooper equalised with three minutes remaining.
The 72-year-old former Scotland boss, who revealed during the week he would be standing down at the end of the season and taking a seat on the Dons board as a non-executive director, said: "It is a very painful defeat. You would have to have seen it to believe it.
"At 3-1 you would have thought a draw was a bad result and it would have been, so a defeat was a very bad result.
"I have been involved in many games but I can't recall a game where you lose a goal in 12 seconds and come back to have control of the game and then you capitulate at the end like that.
"I fault myself in that I should have put on a big striker as a defender.
"But I thought if I did that with Rory Fallon, it sends a negative message to the defence, saying 'I don't trust you' and gives encouragement to Celtic. With hindsight I know what I should have done."