Mitchell Megginson opened the scoring for the visitors, who are managed by one of Easter Road's favourite sons, Ian Murray.
The former Aberdeen youngster blasted into the net from close range following a Scott Agnew cross. It was deja vu minutes later as Mark Gilhaney prodded the ball over the line from another Agnew delivery.
Farid El Alagui was in no mood to concede defeat, however, with the French-Moroccan halving arrears as he rose to glance a fine David Gray cross into the corner of the net.
The same man levelled matters in the closing stages, producing a magnificent header to beat goalkeeper Danny Rodgers before Sam Stanton won it late on.
With the ink barely dry on his one-year contract, Liam Fontaine was immediately pitched into action alongside Jordon Forster.
And the 28-year-old was part of a Hibs side which plumbed the depths then hit the heights during a remarkable evening in Leith. El Alagui and Alex Harris were among those to threaten in a first period dominated by the hosts. It was an encounter which was simmering, rather than igniting, however. That all changed in a remarkable second half.
Dumbarton had offered scarce threat going forward yet claimed the lead on 52 minutes. Colin Nish headed a Scott Agnew corner across the face of goal, and Megginson contorted his body magnificently to volley into the net from point-blank range.
For all their neat football, Hibs were illustrating a complete inability to defend the simplest of set-pieces. A powerful Sons side wasted no time in taking advantage of that. Another Agnew corner caused all sorts of ructions and, after an ugly scramble, Gilhaney forced the ball over the line.
El Alagui halved arrears with 12 minutes left to play, glancing a neat header into the far corner.
Having conceded 11 goals in three games prior to this match, Dumbarton looked vulnerable. And Hibs restored parity with a magnificent El Alagui header, beating Rodgers from all of 14 yards after a splendid cross from Jordon Forster.
There was only one winner now. The only question was whether would it take extra-time to bring it.
Stanton delivered the answer in style, curling a fine left-footed shot past Rodgers from 22 yards.