The holders looked to be heading to extra time for a second consecutive round after a stubborn Inter side defended well.
But Higuain, a constant threat throughout, produced an excellent 93rd-minute finish to score his eighth goal in as many games and set up a last-four clash against Lazio.
Inter, who have now lost three successive matches in all competitions, saw Marcelo Brozovic denied by the woodwork in the early stages of a first half that Napoli controlled.
Higuain had a series of chances in the first 45 minutes, and while the second half was more tame, it produced the all-important goal.
The defeat will heap the pressure on Inter boss Roberto Mancini, recently given a vote of confidence by the club's board.
Mancini, only appointed as Walter Mazzarri's successor in November, sees his Inter side sit 13th in Serie A, and their Coppa Italia elimination means their only realistic chance of silverware this season will come in the UEFA Europa League.
Inter came into the clash without a win at Napoli since 1997, and found life no easier during the opening exchanges this time around.
Higuain first forced a save from goalkeeper Juan Pablo Carrizo after six minutes, before heading narrowly over the bar.
The visitors weathered the early onslaught, however, and were unlucky not to be ahead on 13 minutes when Brozovic, one of eight Inter inclusions, evaded his man on the edge of the box before cracking a low drive back off the foot of Mariano Andujar's right-hand post.
But the hosts continued to threaten and should have taken the lead 12 minutes before the break.
Marek Hamsik's drilled 25-yard strike was palmed into the path of Higuain by Carrizo, only for the Argentina international to poke the rebound straight back at the keeper.
Kalidou Koulibaly then squandered another great chance for Napoli on the stroke of half-time, guiding a volley well over from just six yards.
A cagey opening to the second half saw both sides struggle to create in the final third, with Inter doing well to keep Napoli quiet.
And the seven-time Coppa Italia winners were nearly rewarded for their efforts just after the hour-mark, when George Puscas' angled effort was well kept out by Andujar.
Hernanes saw a goalbound effort deflected wide six minutes later, while at the other end, Napoli, who introduced Dries Mertens and Manolo Gabbiadini in a bid to find a winner, had David Lopez fire well over.
Extra time seemed on the cards but Napoli avoided that when a clever throw-in routine down the left allowed Higuain to shake off Andrea Ranocchia and bend a clinical finish into the top corner to spark scenes of celebration.
