Darren Mackie netted the winner after 17 minutes with a smart turn and finish in the box on Tuesday night.
Dundee, from the Irn-Bru First Division, sent the Dons crashing out of the competition with a shock 3-2 win in 2009, but there was no upset this time around with the visitors unable to find a way past the home defence.
Wales international goalkeeper Jason Brown was handed his Aberdeen debut in place of Colombian David Gonzalez, while midfielder Chris Clark was given his first start in his second spell at the Dons in place of the injured Rob Milsom.
Winger Peter Pawlett and Mackie both started at the expense of Scott Vernon and Youl Mawene. Algerian striker Mohamed Chalali, who had completed his move to Pittodrie earlier in the day, was also named on the Aberdeen bench.
Dundee made only one change from the team that was defeated 4-2 by Livingston, with Ryan Conroy stepping in for Jamie McCluskey.
The Dons made a bright start with Pawlett's long-range drive comfortably held by Rab Douglas before Iceland international Kari Arnason headed over from a Fraser Fyvie corner.
The hosts went closer after 12 minutes when Mackie cut inside, past Dundee skipper Rhys Weston, before curling a left-footed strike inches wide of the target.
The opener for Aberdeen arrived when Pawlett's deep cross was headed into Mackie's path by Clark and the striker turned his marker before hammering the ball into the roof of the net past the despairing Douglas.
The former Celtic goalkeeper prevented his side going two behind a minute later with an excellent stop to keep out Arnason's powerful free-kick.
Dundee rallied and enjoyed a period of concerted pressure before the break, although a Nicky Riley effort which deflected wide and a Stephen O'Donnell shot which was too easy for Brown proved to be the closest they went to an equaliser.
The Dons should have doubled their lead on the stroke of half-time when Clark's tantalising cross found the unmarked Josh Magennis but the Northern Irishman headed over from six yards.
Aberdeen were forced to make a couple of early changes in the second half when Arnason limped off and was replaced by Mawene before Clark picked up a hamstring injury and Chalali was given his debut.
Richard Foster thought he had extended his side's lead after 64 minutes when he reacted quickest to turn the ball home after Douglas had pushed Pawlett's drive against the post, but the goal was ruled out for offside.
Dundee dominated the closing stages and threw everything at the Dons, but the Clydesdale Bank Premier League side survived the onslaught to book their place in the third round of the competition.