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:
5. Jonny Hayes - 8
Snuffed out the attacking threat of Frimpong and posed a threat of his own going forward. Photo: Ross Parker - SNS Group
6. Ross McCrorie - 7
Started at right wing-back before slotting into centre-midfield role following the injury to McGeough. Kept Celtic's creative players quiet for the most part but lost Rogic for the pass which led to Griffiths goal. Photo: Mark Scates - SNS Group
7. Lewis Ferguson - 8
Two excellent penalties - winning one of them himself - took his season's tally to nine and a typically tenacious display in the middle of the park will have only further enchanced his growing reputation. Photo: Craig Williamson - SNS Group
8. Dylan McGeough - 5
Lasted only 25 minutes before being forced off with injury. Worked hard to impose himself on the game in the time he was on. Photo: Craig Williamson - SNS Group