Andrea Pirlo's side came into the tie with a 2-1 advantage from the first leg in Milan on the back of a Cristiano Ronaldo brace, as the Serie A holders looked to book a spot in the final.
Ronaldo had two chances late in the first half, but saw both turned away by the Inter defence and goalkeeper Samir Handanovic as the teams went to the dressing room locked in a goalless draw.
Handanovic was called into action again shortly before the 70-minute mark to palm away a powerful Ronaldo shot that could have put the tie away for Juventus.
Inter threw everything at Juve late on, but veteran keeper Gianluigi Buffon and his defence stood firm to secure a spot in the showpiece match.
"We deserved it having taken on a great team like Inter," Pirlo said.
"In such important matches our team is unlikely to make mistakes, they all played a great match. We changed something on a tactical level and they did well, we had the clearest opportunities."
With the away goals rule in place, Inter had to go for it and, with Lautaro Martinez and Romelu Lukaku in the starting lineup, Antonio Conte's side certainly had the firepower to trouble the hosts.
But 43-year-old Buffon, used only sparingly by Juve boss Pirlo this season, was rarely tested, despite Christian Eriksen putting plenty of dangerous crosses into the box.
Inter committed more players forward as the match wore on, but the pressure was soaked up by the home defence as the visitors finished the match without scoring for the seventh successive trip to Juventus in all competitions.
"I think Inter tonight showed we have narrowed the gap to Juventus," Conte said.
"The ambition must be high, then the reality is something else and reality says that we are out. We have been knocked out of the cup against the team that has been winning [Serie A titles] for nine years and I think we deserved more."
The result booked the Turin side a 20th Coppa Italia final appearance on May 19, with a re-run of last season's showpiece -- when Napoli beat Juve on penalties -- still possible.