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:
9. Ryan Hedges - 8
A goal was just reward for a performance full of running power. Was a handful for the Celtic defence throughout and runs Ferguson close for Aberdeen's star performer. Photo: Bill Murray - SNS Group
10. Marley Watkins - 6
Produced a deft through ball which led to the penalty opener and grafted throughout his 56 minutes on the park. Photo: Ross MacDonald - SNS Group
11. Ryan Edmundson - 5
The quieter of Aberdeen's front men, struggled to get into the game before being thumped into the advertising boards by Duffy leading to his 57th minute withdrawal. Photo: Craig Williamson - SNS Group
12. Connor McLennan - 6
Replaced the injured McGeough in the first half. Struggled to get forward until winning the penalty which rescued a point for Aberdeen in injury-time with a neat turn drawing a foul from McGregor. Photo: Craig Williamson - SNS Group