Rodrigo Palacio and Antonio Cassano led Andrea Stramaccioni's team to a remarkable comeback in Milan but Emmanuel Adebayor struck to send Andre Villas-Boas's men through
Antonio Cassano and Rodrigo Palacio put a serious dent in Spurs' 3-0 first-leg lead before the former's free kick deflected in off William Gallas with a quarter of an hour to play.
After a disastrous 90 minutes, however, Spurs responded well in extra time and Adebayor tapped in from close range after Mousa Dembele's shot was parried into his path.
Ricky Alvarez headed home to make for a tense finish but Inter could not find the fifth goal that they needed to claim a place in the last eight.
Andre Villas-Boas's team defended high up the pitch and attempted to take the sting out of any early Inter momentum by dominating possession in midfield, which they did while asking few questions of Samir Handanovic.
Cassano and Palacio, though, were always likely to take advantage of that aggressive back line given the chance and the former had tested Brad Friedel with a weak effort before he tumbled in the box from a similar position under very little contact from Gallas.
The referee was unconvinced on that occasion but Palacio was more purposeful when played in by Jonathan on the right and crossed for Cassano to head home at the back post.
Inter attacked with renewed vigour having cut into Tottenham's aggregate lead and the experienced Italy striker almost repaid the favour by setting Palacio through on goal but his chipped effort hit the crossbar.
Spurs briefly stymied the home side's flow more effectively thereafter and Sigurdsson saw his curling shot tipped around the post but Inter would threaten again before the half was out.
Palacio's cross just evaded Esteban Cambiasso and Fredy Guarin latched on to yet another pass lobbed behind the Tottenham defence before cutting back and shooting powerfully at Friedel, who parried.
Adebayor could have put the tie to bed against the run of play with the last kick of the half, striding through on goal after industrious work from Jermain Defoe had caused confusion at the back, but his attempted chip flew over the bar.
Adebayor remained a handful after the break, with one centre after an imperious run down the left wing just diverted away from Jan Vertonghen by Christian Chivu, but the game was becoming increasingly stretched as the two sides traded forward thrusts.
And after Dembele had conceded possession to Cambiasso in midfield, the Argentine had plenty of time with which to pick out his compatriot Palacio, who slotted past Friedel.
That error aside, Dembele's move into central midfield after the introduction of Lewis Holtby for Defoe helped Spurs to tighten their grip on midfield until Inter levelled the tie in fortuitous circumstances.
Cassano's low free kick from the left looked to be heading wide after deflecting off the wall until Gallas ill-advisedly stuck out his leg to redirect it into the bottom corner.
Handanovic was called into action when Kyle Naughton rocketed a 25-yard shot across goal before Cambiasso wasted a chance to win it when one-on-one with Friedel after a poor sliding tackle by Gallas.
Villas-Boas introduced Lennon shortly before the end of the 90 minutes and, with the prospect of one Spurs goal leaving Inter needing to take their tally to five, the indefatigable Javier Zanetti denied the winger a shot on goal in the opening stages of extra time with an excellent sliding challenge.
From the resulting corner, Handanovic saved brilliantly from Vertonghen before Gallas sent a free header wide of the post from the next delivery.
Spurs had started the first half of the additional 30 minutes firmly on the front foot, though, and left Inter with a mountain to climb when Handanovic could only turn Dembele's shot into the path of Adebayor, who slid in to convert a crucial away goal from close range.
Inter refused to lie down and, when Cassano picked out Alvarez from the left, the substitute nodded in to give Andrea Stramaccioni's team hope but Spurs survived the inevitable late onslaught.