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.
"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."
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 minute remaining. That left the stage set for Samaras, who had replaced Beram Kayal, to hook in the winner with seconds left.
Lennon added: "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."
Samaras claimed it was the "most exciting victory" since he joined the Parkhead club in 2008.
"It was a great goal for the team," said the Greek striker. "We showed great character to come back, it is something special so well done to all the boys.
"Sometimes we play bad and lose, sometimes we play bad and win, but nothing can compare with this game. I remember Jan Vennegoor of Hesselink scored in the last minute once but I can't remember another game when we had this type of comeback."