The Red Army did their part, but a stuttering Dons had a get-out-of-jail and avoided a replay at Dumbarton, thanks to a penalty award five minutes from full time.
Aberdeen had been held at home at this stage by Stenhousemuir 12 months ago, and another replay looked likely.
Derek McInnes made five changes to the side that drew 1-1 by Hearts in the final game of 2019.
Dylan McGeouch made his debut for the Dons after arriving from Sunderland.
Sam Cosgrove following his two-game suspension and Andrew Considine was back from injury.
Also returned to the starting line-up was Niall McGinn and Connor McLennan.
They covered for the injured Zak Vyner, Lewis Ferguson was suspended. Dean Campbell, Jon Gallagher and Curtis Main were all benched.
Aberdeen had more than enough shots at goal, but couldn’t make any of them count. 28 in total.
And the first chance came from new signing Dylan McGeouch, who fired a shot just inside the box, but it was straight at Conor Brennan, who comfortably saved.
Shortly after, Niall McGinn fired in a corner kick that dropped to Ash Taylor, but he flicked the header from 12 yards over the bar.
Bruce Andersons should have put the Dons ahead in the 27th minute, when a shot from Cosgrove was blocked and fell to Anderson, but his shot flew over the bar.
Andy Considine was next up to have a go, but he headed over the shot from a McGinn corner.
McGinn also had a chance, but his long range effort went over goal.
Anderson had another go, but he hesitated with his shot and when he did it was too late and deflected out for a corner. From the corner. Andy Considine headed the shot straight at the keeper.
The Dons continued with their chances in the second half. Niall McGinn shot across the face of goal but the shot went wide in the 47th minute.
Cosgrove headed over from 12 yards out in the 53rd minute from a McGinn cross into the penalty area and moments later Scott McKenna headed over from a McGinn corner.
McGinn picked out Anderson in the 63rd minute but he fired the shot over on the run from 12 yards.
In the 73rd minute, Anderson fired a powerful drive from 15 yards, but Brennan saved the shot.
McGinn had his final chance of the match when he curled an effort wide of goal in the 80th minute.
Aberdeen were awarded a penalty in the 85th minute after Sam Cosgrove was taken down in the box by Rico Quitongo. Sam comfortably fired the ball into the net putting Brennan the wrong way to send the Dons through.
It was harsh on Jim Duffy’s side, but the Dons are through to the next round – but any chance of a Scottish Cup Final for the Dons will need a much improved performance.
Derek McInnes told the club website: “I think it was a combination of the Dumbarton lads being very structured and defending well and us not taking our chances.
“I think we had 28 shots, almost 70% possession, those two thirds of the pitch were very good for us in terms of dominating the game and controlling things.
“I thought McGeouch and Ojo in particular were very strong in there against three midfielders. We went with two up front with two wingers to try and stretch them and have that threat in the box, we never managed to capitalise on that.
“I am still bemoaning the fact that we passed up so many chances. From set plays everyone of our bigger lads had an opportunity; Considine, Taylor, Cosgrove, McKenna. They have got to be better than that.
“Every time that we missed a chance it is a shot of adrenaline for the Dumbarton boys. The longer you don’t score, they start to believe it will be their day. The longer that we don’t score, we maybe start to think it is heading for a replay.
“I am delighted that the players managed to keep their composure.
“Big Sam doesn’t snatch at this chance. He has got the calmness. He has nice quick feet in there to win the penalty.
“He didn’t have the best of afternoons, but he is still the guy who makes the difference for us.
“He anticipates the challenge coming and sticks away the penalty.
“Nobody can deny that we deserved to win the game. It was the type of performance that maybe merited more goals, but you have still got to score them.
“For now, it was about getting through the tie. We deserved to win, we deserved to win by more goals but it was a victory nonetheless.
“We are searching for that bit of penetration and quality in the final third.
“We tried to be as attacking as possible going with two strikers and two wide players, it didn’t make any difference to the score line and it is something we need to improve on.
“It is a part of their game that is very important. These boys have got to take more ownership to score more goals. When the delivery is as good as it was from McGinn today, it was on the money every time. Everyone of them had a chance from a set play.
“I think we had umpteen corners, free-kicks going into the box.
“Eventually if you take your chances, even from set plays, Dumbarton need to come out to get something from the game, then there might be more opportunities.
“The longer it is 0-0, it went on long enough being 0-0, the more they think they are in the game.
“I thought the performance was more than decent, just that last bit we need to sharpen up our work.
He also commented on The Red Shed: “I thought it was magnificent, absolutely brilliant.
“They came here determined to put on a show and they did exactly that.
“One of my players commented on how different the atmosphere felt.
“I think if that wasn’t there today, that could have been a more difficult game.
“They stuck with the team, they kept behind them.
“It felt like an away support and we get brilliant backing from our away fans.
“I cannot thank everyone enough who turned up at the game today. The ones in the Red Shed, I thought they got behind the team brilliantly.
“The atmosphere felt brilliant from the warmup. If that is a sign of things to come, then long may it continue.
“We would have loved to have given them more goals, we would have loved to have put a bit more polish on the performance for them but if they can continue to do that I have no doubts that they will help the team.
“Hopefully the supporters enjoy being a part of an atmosphere like that as well.
“We are all a bit frustrated we haven’t scored more goals but there is no doubting it generated a far better atmosphere and full marks to them for that.