Darren Mackie's first goal in over a year sent Aberdeen into the third round of the Scottish Communities League Cup at the expense of Dundee.
The striker, who returned to action this season after an eight-month injury lay-off, converted Chris Clark's knock-down on 16 minutes.
Peter Pawlett, Kari Arnason and Josh Magennis also threatened for the hosts.
Dundee improved after the break and Stephen O'Donnell was twice denied by Aberdeen keeper Jason Brown late on.
Aberdeen started brightly and Pawlett had an effort saved by goalkeeper Rab Douglas before Arnason's header cleared the Dundee crossbar.
The breakthrough came when Clark headed Pawlett's cross towards Mackie, who turned his man well before hammering the ball into the roof of the net.
Dundee pushed upfield in search of an equaliser but Nicky Riley's effort was deflected wide and O'Donnell's shot was comfortably saved by Welsh debutant Brown.
Arnason had a free kick saved by Douglas and Josh Magennis headed over as Aberdeen continued to press.
Arnason and Clark picked up injuries in the second half and were substituted by Youl Mawene and new signing Mohamed Chalali respectively.
Aberdeen captain Richard Foster had a goal disallowed for offside and also fired wide while O'Donnell had two efforts saved by Brown in the closing stages.
The Dons boss Craig Brown admitted at full-time that he was relieved to come away with the victory.
"The match was a bit similar to our game on Saturday against Inverness where we were good at the beginning and a bit indifferent thereafter," he said.
"If you are highly critical of Aberdeen then I think it would be disrespectful of the opposition, who were excellent.
"They would have given anybody a good game tonight."
Meanwhile the Dundee manager Barry Smith thought his side equipped themselves well against top-flight opposition.
"I'm obviously disappointed to get beat but I'm not disappointed in the manner that we got beat," he said.
"I think we gave a very good account of ourselves against a Premier League team."