Colin Nish, their veteran striker, contributed the only goal and Dumbarton passed up a number of chances to make the tie more comfortable.
Dumbarton put Alloa under early pressure and Jordan Kirkpatrick sent a couple of efforts wide before Scott Agnew was also off target from the edge of the area.
Nish scored the all-important goal after 31 minutes with a close-in finish after Scott Agnew had picked him out in the area.
Dumbarton were then awarded a penalty when Chris Kane tumbled after being brought down by Scott Bain, but the goalkeeper made amends by saving Kane's kick.
Bain was injured in the process and was replaced at half-time by Craig McDowall.
Kane had a good chance for a second goal midway through the second half but could not connect properly in front of goal.
Scott Linton was unfortunate to see a shot come back off the crossbar as Alloa were hanging on at times.
Kevin Cawley missed a great chance for an equaliser but mistimed his header in front of goal as Dumbarton held on.
Albion Rovers reached the quarter-finals for the first time in 80 years with a 2-0 home win over Stenhousemuir.
The Lanarkshire side, giant-killers following their slaying of Motherwell in the previous round, will now be hoping for a big-name tie in today's quarter-final draw.
Darren Smith came close with an early header for Stenhousemuir, but it was Rovers who drew first blood.
The home side moved in front midway through the first half when Ross McMillan misjudged a high, through ball in the wind and headed the ball over his own goalkeeper Chris Smith.
Ciaran Donnelly almost made it 2-0 before half-time, but Smith was well positioned to save.
It was left to Gary Phillips to double Rovers' advantage with a well-taken goal midway through the second half and Stenhousemuir could not muster anything to save the tie.