It had finished 1-1 after extra-time after Ciro Immobile linked up with Felipe Caicedo to put Lazio ahead. But Mauro Icardi’s 125th-minute penalty for Inter took the game to a shootout.
Lazio’s Riza Durmisi had his effort saved by Samir Handanovic, but Immobile, Marco Parolo, Franceso Acerbi and Lucas all scored to take Lazio into the final four.
The semi-finals are played over two legs – on 27 February and 24 April – with the final taking place at Lazio’s home ground, Stadio Olimpico, in Rome on 15 May.
Handanovic had twice denied Immobile and also stopped a shot from Caicedo in normal time, with Antonio Candreva missing Inter’s best opportunity when he shot over from six yards.
In a frantic extra-time period, Immobile put his side ahead, before Inter thought they had Kwadwo Asamoah sent off for fouling Sergej Milinkovic-Savic, but the decision was downgraded to a yellow card after a video review.
Then, with the last kick of extra time, Icardi’s penalty, awarded after Milinkovic-Savic’s foul on Danilo D’Ambrosio, took the game to a shootout, with Lazio’s Stefan Radu dismissed for arguing that decision.
Cedric Soares made his Inter debut following his loan move from Southampton on Saturday and scored one of his side’s penalties, but it was to be a losing debut for the defender.