A brace of penalties from Lewis Ferguson, the second of which rescued a point in injury time, either side of a Ryan Hedges goal cancelled out strikes from Callum McGregor, Leigh Griffiths and a Ryan Christie penalty for the defending champions.
The two sides meet again in next Sunday's rearranged Scottish Cup semi-final from the 2019-20 campaign.
Here's how the Aberdeen players rated:
1. Joe Lewis - 6
Had a quiet, albeit assured, first half. Busier after the break, making saves you would expect him to make and had no chance with Celtic's frst two goals before being sent the wrong way from the spot
Photo: SNS Group
2. Tommie Hoban - 6
Mistimed tackle in the box on Elyounoussi blighted an otherwise competent showing on the right of Aberdeen's back three.
Photo: Ross MacDonald - SNS Group
3. Ash Taylor - 6
Kept starting striker Ajeti quiet but was side-stepped too easily by McGregor for Celtic's first then gave Griffiths too much space in the box for the second.
Photo: SNS Group - Craig Williamson
4. Andrew Considine - 6
Strolled the first half but, like his defensive colleagues, found himself under more pressure after the break. Remained assured throughout.
Photo: Ross Parker - SNS Group