Dons boss Derek McInnes hailed his gritty side for finding a way into the Betfred Cup quarter-final when not at their best.
Aberdeen won 2-1 against Dundee after extra-time at Dens Park in the last-16 of the League Cup.
After the exertions of playing for 70 minutes with 10 men against Rijeka in the Europa League on Thursday the Reds struggled at times in the City of Discovery.
Danny Johnson fired Dee ahead with a penalty just before half-time, but Andy Considine equalised in injury-time to force an extra 30 minutes in which Sam Cosgrove netted the winner.
In the quarter-final the Dons will play Hearts at Tynecastle on September 24, 25 or 26.
After their comeback, McInnes said: “I need to applaud Dundee. I watched them last Saturday and not only did we have to deal with a team motivated, playing with confidence and who had prepared all week, they were certainly ready for it. We turned up for the game and made the players aware of how difficult it would be, especially after playing with 10 men for so long on Thursday.
“We were not brilliant. It was all about digging deep and getting the result.
“Two weeks ago we left Pittodrie after beating Hearts 3-2 and everyone was euphoric.
“Saying ‘this was the best team we had had in a while’ and we cannot go as high as that to as low as we were.
“Dundee gave us a right game and made it difficult for us. As you know cup games are all about finding a way to win and my players deserve so much praise for that.
“We’ll give them a few days off and then get back to some training days which we haven’t had the luxury of in recent weeks.
“It takes luck, good players and quality but in any team there has to be a strength to dig out results.
“For long spells it didn’t look like there was a goal in us but we made changes, took a risk by going man for man at the back and thankfully we had enough bodies at both goals to get the reward.
“A lot of people would have taken pleasure in seeing us go out of the cup and I have a lot of pleasure in seeing my team go through.
“I know we can improve, be far more slicker and more technical, and I know we’ll have more energy in our legs.
“There are certain games where you need grit and determination and we had that.”
Considine conceded the penalty for a foul on Jordon Forster, but redeemed himself with the equaliser.
McInnes added: “If Andy has conceded a penalty he has certainly made up for it.
“It was a real dagger to the hearts of the Dundee players late on and we looked more equipped to get the second goal and once we got it we managed to get over the line without too much fuss.”
McInnes’ celebrations for both his team’s goals were a mixture of relief and jubilation, in many ways mirroring the emotions of the Red Army who had travelled to Dens.
He said: “Our fans were put through the mill. They came here with high expectations and it wasn’t playing out as they expected.
“We have to credit Dundee for that. You just become like a supporter.
“You just feel that elation and relief.
“It was relief and sheer pleasure on my part at seeing my players having to dig deep and winning.
“Funso Ojo, Shay Logan, Greg Leigh, Scott McKenna and Andy Considine had nothing left at the end but they were still putting everything in.”
Dundee boss James McPake said: “I’m proud but I expect my team to be like that.
“That type of performance has been coming for a while but I’m devastated for them and the fans.”