Inter, fresh from six successive victories prior to Wednesday's (AEDT) game, took the lead in magnificent fashion through Jeison Murillo, who produced a finish belying that of a centre half to convert Joao Mario's corner with a sensational overhead kick.
Joao Mario then combined with Rodrigo Palacio to make it 2-0, but Bologna was given hope three minutes before half-time courtesy of Blerim Dzemaili's deflected effort.
Stefano Pioli's men had numerous opportunities to put the game to bed in the second half and eventually paid the price for not taking them when Godfred Donsah headed Bologna level.
A decisive goal did not arrive inside 90 minutes, but in the first period of extra time Inter benefited from a deflection of their own as Candreva struck what proved to be the winner.
Inter will now face the winner of Thursday's contest between Candreva's old team Lazio and Genoa.