Ciro Immobile's scuffed and deflected strike eluded the otherwise-brilliant Samir Handanovic and looked set to secure Lazio's progress until Mauro Icardi levelled with a penalty with seconds remaining.
Lucas Leiva struck what proved to be the winning spot-kick after former Roma man Radja Nainggolan saw his effort saved, ensuring another quarter-final triumph for Lazio and placing more pressure on Inter boss Luciano Spalletti.
Handanovic twice denied Luis Alberto before he produced a stunning reaction stop to deny Senad Lulic, although the offside flag would have disallowed any goal.
Inter then wasted the best chance of the first hour, as Antonio Candreva, against his former club, somehow side-footed high and wide with the goal gaping after Thomas Strakosha parried Joao Mario's shot into his path.
Felipe Caicedo had the chance to score the winner off the bench with barely two minutes left of normal time, but Handanovic again made a superb save to keep out the striker's uncertain finish, before tipping another Immobile shot wide.
In injury time, Strakosha blocked a Matteo Politano shot and the loose ball fell to Lautaro Martinez, who slipped and scuffed the shot over the bar.
5 - Five of Inter's last seven Coppa Italia games went into extra-time. Routine. #InterLazio— OptaPaolo (@OptaPaolo) January 31, 2019
Mauro Icardi saw a penalty shout waved away after a push by Sergej Milinkovic-Savic as Inter began to look likeliest to strike in extra time, but it was Immobile who at last broke the deadlock and sent Lazio through, the striker combining with Caicedo and sending a deflected shot just beyond the reach of Handanovic.
Kwadwo Asamoah had a late red card reduced to a yellow after a VAR review, but the bigger drama was yet to come, with Milinkovic-Savic ruled to have fouled Danilo D'Ambrosio by the video assistant in the last minute of stoppage time, allowing Icardi to bury an equaliser with the final kick after Stefan Radu had been sent off for his appeals.
Martinez and Riza Durmisi had penalties saved, but when Strakosha stood tall and parried Nainggolan's effort over the bar, Lucas swept home to send his side into the next round.
What does it mean? Lazio to face AC Milan
Simone Inzaghi's side will meet Milan in the last four, the Rossoneri having impressively overcome Napoli in their quarter-final on Wednesday.
Inter will consider this a good opportunity missed for a real chance of silverware, but their focus will now return to Champions League qualification.
Handanovic almost unbreachable
Handanovic made eight saves in normal time alone in what was a quite brilliant display of shot-stopping. His stop from Durmisi in the shoot-out was outstanding.
Candreva proves costly
Candreva did little to threaten his old club until the 65th minute, when he contrived to miss an open goal. He was probably fortunate to stay on the pitch, given his level of impact.