Inter, beaten 3-0 in the first leg in Turin, rallied brilliantly to win the return match by the same score with two goals from Marcelo Brozovic and another by Ivan Perisic.
Inter's Rodrigo Palacio had the only miss of the shootout, the forward's effort bouncing back off the bar. Leonardo Bonucci's kick sent Juve through to the final.
"That was the worst performance we’ve had since I’ve been at Juventus,” Bonucci told Rai Sport. “We got the approach wrong and made things easier for Inter.
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“We made life difficult for ourselves and were fortunate that it went it penalties.”
A weakened Juve were lethargic and unadventurous and the visitors were punished for their complacency as Inter dominated the second leg.
Goalkeeper Neto, standing in for Gianluigi Buffon, handed Inter their opening goal when his clearance was charged down by Gary Medel.
The Chilean midfielder, nicknamed 'Pit Bull', appeared to barge Hernanes off the ball before playing in Brozovic, who finished emphatically in the 17th minute.
Adam Ljalic almost reduced Inter's deficit further when he struck the bar but the hosts finally made it 2-0 on the night in the 49th minute when Eder's cross from the right was finished by Perisic.
Neto was again at fault for the third goal in the 82nd minute as he let Brozovic's penalty go under his body after Daniele Rugani had brought down Perisic.
"It was a wonderful game,” Inter coach Roberto Mancini told the Italian media, “I compliment the players for how they managed it.
“I am sad for them, they deserved to go to the final.”*
Milan, who defeated third tier Alessandria 6-0 in their semi-final on Tuesday, will now meet Juve in the final for the first time since 1990.
(Writing by Ed Dove; Editing by Tony Jimenez/Peter Rutherford)
