The Serie A giants wiped out the Londoners' 3-0 advantage from the first leg with goals from Antonio Cassano, Rodrigo Palacio and an own-goal from William Gallas.
Emmanuel Adebayor scored in the 96th, only for Ricardo Alvarez to give the Italians hope again in the second period of extra-time, but Andre Villas-Boas' side held on to advance.
The match action was a thrilling advert for the often-derided competition, but the night was tarnished by racist abuse from the home fans towards Adebayor.
Inter started like a team possessed. Cassano beat Brad Friedel at the third time of asking after two earlier strikes. The 30-year-old got on the end of a Palacio cross after Jonathan cut out Kyle Naughton and headed home from six yards.
Tottenham held back the tide before the interval, but soon after the restart Cassano found Palacio with a through-ball and he curled a shot beyond Friedel to make it 2-0.
Inter's quest to find a third looked set for failure until they won a free-kick 25 yards out with 15 minutes to go. Cassano struck the wall with a powerful low effort but it squirmed through and Gallas stuck out a leg and poked the ball past an off-balance Friedel.
Villas-Boas gathered his team together after the referee signalled the end of the normal 90 minutes and Tottenham seemed reinvigorated by the talk from their manager. Handanovic saved well from Aaron Lennon and Jan Vertonghen, while Gallas headed wide unmarked from eight yards.
Adebayor did not miss the target moments later, however, stabbing past Handanovic after the Inter goalkeeper spilled Mousa Dembele's shot.
The 2,000 travelling fans erupted in ecstasy, but they were back biting their finger nails again in the 110th minute when Ricardo Alvarez nodded home from Cassano's cross to make it 4-4 on aggregate.
The away goal meant Tottenham remained on course to go through, and despite late attempts from the home side Villas-Boas' team held on to enter the hat for the quarter-finals.
|Internazionale||Team Statistics||Tottenham Hotspur|
|1||1st Half Goals||0|
|12||Shots on Target||5|
|7||Shots off Target||4|
|44.7||1st Half Poss.||55.3|
|47.5||2nd Half Poss.||52.5|