Trailing to second half goals from Mitchel Megginson and Mark Gilhaney, Hibs responded in the last 12 minutes with a brace from Farid El Alagui and a winner in the dying moments from Sam Stanton.
Hibs applied plenty of pressure in the first half but could not find a way through as Alex Harris had his shot cleared off the line, El Alagui's low strike was well saved and no-one was able to get on the end of a superb David Gray cross.
Dumbarton rallied after the break and went ahead on 52 minutes when Mitchel Megginson connected with a cross from the right and his acrobatic shot ended up in the net.
Seven minutes later and Dumbarton looked to be on their way to the next round when Gilhaney scrambled the ball home over the line from a set-piece.
El Alagui pulled a goal back for Hibs with a header i and the striker bagged an equaliser six minutes later.
Stanton completed the comeback when he crashed a thunderous shot in the closing stages to cap a memorable night for the Leith side.