The striker scored in the 90th minute for Paul Hartley's home side at Dens Park to send Derek McInnes' men out of the competition. Both sides looked to have settled for a replay at Pittodrie next month before in-form September signing Clarkson scored his eighth goal for Dundee to clinch a 2-1 win.
German defender Thomas Konrad had headed the home team into an early lead, but it was also he who turned Niall McGinn's cross into his own net for Aberdeen's 17th-minute equaliser.
For the fifth-minute opener, Gary Harkins' free-kick picked out Konrad who powered a header down and past visiting goalkeeper Scott Brown.
The Dons levelled against the run of play when Jonny Hayes went on a driving run before releasing McGinn who had made a strong run on the overlap. McGinn had timed his run to perfection and the Northern Ireland international dug out a cross which hit Konrad and rolled into the net past helpless Scott Bain in the home goal.
Some of Dundee's impetus started to go as Aberdeen strived to get in front, with their next chance coming just after the half-hour mark. Hayes was involved by forging down the left before digging out a deep cross, but David Goodwillie could not quite get his shot away at the far post and the hosts breathed a sigh of relief.
Dundee had a golden opportunity to regain the lead after 37 minutes when Martin Boyle's cross was missed by Dons defender Shay Logan, but Greg Stewart hacked his shot over the bar when he had more time than he thought.
At the other end, Dundee had goalkeeper Bain to thank for keeping them level soon afterwards. Goodwillie's cross from the right was met by Irishman Adam Rooney whose powerful downward header looked net-bound, until Bain somehow got down to palm it away with a wonderful stop low to his right.
The second half started at a slightly lower tempo although Aberdeen's Ryan Jack attempted to lift the game with a long-range effort on the hour mark which Bain had to parry at the expense of a corner.
Dundee made two substitutions midway through the half with Simon Ferry replacing Harkins, and Willie Dyer taking over from McGinn.
After 77 minutes it was Aberdeen who almost struck to win the tie. Hayes' jinking run into the box ended with him cutting the ball back for Rooney, but with the goal at his mercy he shot high into the side-netting. Hayes then forced another save from Bain from a free-kick as Aberdeen turned the screw.
Instead Clarkson sealed the victory with a cool 12-yard finish beyond Brown, after James McPake headed down Gary Irvine's ball forward.
But there was still time for drama in the final seconds as Niall McGinn couldn't have come closer to levelling. The Dons forward smashed a close-range effort towards goal and the excellent Bain got the slightest of touches deflecting the shot onto the bar where it bounced down onto the goal-line and up onto the bar again before rebounding to safety.