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:
13. Scott Wright - 8
A teriffic impact off the bench. Sublime dribbling ran the Celtic defence ragged in the last half hour and key involvement in both Aberdeen's second half goals. Photo: Craig Foy - SNS Group
14. Sam Cosgrove - 7
Off the bench for his first appearance of the season and made his presence felt with an assist on Aberdeen's second goal and also saw a shot come off the underside of the bar prior to the last minute penalty award. Photo: SNS Group