Adam Rooney scored the only goal at Pittodrie eight minutes into the second half to set up a semi-final at Ibrox next month.
Dumbarton were enjoying a decent spell in the match when the Dons scored the decisive goal as Rooney powered his header into the net from a Barry Robson corner.
Robson should have made more of an early chance when he ghosted in at the back post to meet a Jonny Hayes cross, but headed the ball straight at Steve Grindlay.
And Robson tested Grindlay two minutes later when he wriggled into space at the corner of the penalty area and curled a shot towards the far post which forced the Dumbarton keeper to turn the ball behind for a corner.
But Russell Anderson came to Aberdeen's rescue shortly before the opening goal when he cleared after goalkeeper Jamie Langfield spilled the ball through his legs, with Chris Kane just about to pounce.
Anderson then came close to doubling the advantage when he darted in at the near post to glance a header narrowly wide.
Two minutes later, Hayes almost sneaked a header in at the back stick before Ryan Jack had a deflected shot bobble off the upright as the Dons tried to kill the tie.
But Dumbarton almost secured a replay when a deep Garry Fleming cross found Paul McGinn and his downward header cleared the bar with just two minutes remaining.