An injury-time strike from substitute Georgios Samaras took the champions 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.
That left the stage set for Samaras, who had replaced Beram Kayal, to hook in the winner with seconds left.
"It was absolutely brilliant," said Celtic manager Lennon. "People were accusing the players or criticising the players for complacency and not showing enough desire, but when the fourth goal went in I think you could see what it meant to them and what it meant to the supporters.
"It was like the good old days. I know we have got a few defensive problems that we need to sort out. We keep giving cheap goals away but our desire to win the game was fantastic.
"I have been waiting on this team to do that for a wee while and they showed me that here."
- Sports & Recreation
- Georgios Samaras