Massimiliano Allegri’s side trailed 2-1 going into Tuesday’s game but levelled on aggregate when Alessandro Matri fired home a rebound.
Roberto Pereyra poked in a second before Leonardo Bonucci volleyed in.
Juventus, who had Alvaro Morata sent off late on, will face either Napoli or Lazio in the final.
Juve are looking to complete a treble this season.
They have a 14-point lead at the top of Serie A and are in the Champions League quarter-finals where they face AS Monaco.
Juventus’ win on Tuesday came despite the absence of the injured Carlos Tevez, Andrea Pirlo, Paul Pogba and Stephan Lichtsteiner. They are looking to win the Coppa Italia for the first time since 1995. —