This shock defeat leaves the Bhoys just two points ahead of second placed Rangers.
Aberdeen boss Jimmy Calderwood said: "It was a fabulous advertisement for Scottish football and it is a massive boost for us to beat the champions.
"We took a lot of flak round about September/October when we were bottom of the league.
"But we've got the youngest team in the SPL with a lot of new boys and, all credit to them, they have done wonderfully well to turn it around.
"I thought we deserved to win the game, we are now in third and there's no reason why we can't keep the momentum going."
But Dons defender Zander Diamond struck twice in three second half minutes to give Aberdeen a sixth successive home victory.
Celtic boss Gordon Strachan tried to remain positive.
"The consolation is that my team played well, made chances, showed great determination and great character, while the two goals we scored were absolutely fantastic.
"If they play like that for the rest of the season, I will be happy. If they defend like that at set-plays, I will be a worried man.
"The thing that let us down is that they scored four goals from four free-kicks and we have to see if there is a common denominator there and eradicate it.
"I feel sorry for Mark Brown, because he has not had a save to make and has not been able to show his ability."