The young Hibees fell behind on Tayside to Thomas Bradbury’s 25th-minute header following a John Lambert cross from the right.
However, Grant Murray’s side were only behind for a matter of minutes as Fraser Murray drilled his effort into the back of the net.
Hibs seized the initiative after the break through full-back Sean Mackie’s powerful shot from inside the penalty box with little more than 15 minutes left to play.
However, Dundee weren’t to be denied and they restored parity in the 85th minute through a Daniel Jefferies header as the tie went to a penalty shootout.
Despite the hosts missing the first spot kick of the evening, Innes Murray and Ben Stirling spurned their kicks from 12 yards and Bradbury kept his cool to send Dundee through.