Emmanuel Adebayor answered the racist taunts of the home fans by scoring a killer away goal on a night that so nearly ended in humiliation for Andre Villas-Boas' shellshocked side, who crushed the Italian giants 3-0 in the first leg but were completely outclassed in the first 90 minutes on Thursday.
Antonio Cassano put Inter ahead after 20 minutes and Rodrigo Palacio made it 2-0 after the break before William Gallas stabbed the ball into his own net to level the tie at 3-3 on aggregate.
Inter laid siege to the Spurs goal in the hunt for a killer fourth, but the visitors held on and Adebayor scored against the run of play in the first period of extra-time.
Inter scored again through Ricardo Alvarez to bring the tie level on aggregate, but Adebayor's away goal was enough to see Spurs through to the last eight of the Europa League.
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.
The Togo striker, who has suffered racist abuse in the past, was the target of monkey chanting.
UEFA will surely come under pressure to punish Inter, whose fans also waved inflatable bananas at Mario Balotelli in last month's Milan derby.
Villas-Boas' team, without suspended star man Gareth Bale, dominated the early exchanges, but they did not test goalkeeper Samir Handanovic.
Jan Vertonghen swung his right leg at Gylfi Sigurdsson's cross, but he missed. Thereafter Inter took control.
Esteban Cambiasso picked out Cassano with a through-ball, but Brad Friedel saved well from the Italy striker.
Cassano broke through the Spurs back four once again after William Gallas misjudged the flight of the ball. The Frenchman raced back and appeared to nudge Cassano over, but the referee did not award a penalty.
Cassano did not fail to beat Friedel at the third time of asking, however. The 30-year-old got on the end of a Palacio cross after Jonathan cut out Kyle Naughton and headed home from six yards.
A flare went off in the home end and Inter's fans found their voice. Villas-Boas sprung to his feet and began ordering his team to remain calm.
Inter did not relent, however. They almost added a second goal in the 26th minute when Cassano threaded a ball through Kyle Walker and Gallas to find Palacio, but the Argentinian's lob cannoned off the bar and over.
Tottenham were panicking all over the pitch. A mix-up between Gallas and Friedel almost allowed Palacio in before Cassano shot just wide of the American's goal.
Tottenham still offered little going forward. Walker glanced a header wide. Jake Livermore and Friedel were booked for time-wasting.
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.
Villas-Boas threw on Lewis Holtby and Aaron Lennon in a bid to give his team more impetus, but Inter continued to lay siege to the away box.
The evergreen Javier Zanetti entered the Tottenham area and fired a low shot at Friedel's near post, but he blocked the strike and Tottenham's fans breathed again.
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.
Two minutes into injury time Gallas missed a horrible challenge on Cambiasso, but the Argentinian shot wide from 10 yards.
Villas-Boas gathered his team together after the referee signalled led 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.