Spurs needed an away goal to progress after losing 4-1 on the night, with Adebayor scoring with an outstretched boot in extra-time.
Tottenham crushed Inter 3-0 in the first leg, but the Italian giants completely outclassed Andre Villas-Boas' shell-shocked side in the first 90 minutes.
Antonio Cassano put Inter ahead after 20 minutes and Rodrigo Palacio made it 2-0 after the break before William Gallas stabbed the ball in to 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 they held on and Adebayor scored against the run of play in the first period of extra-time.
Ten-man Basel produced a desperate defensive display to book their place in the quarter-finals of the Europa League at the expense of a wasteful Zenit St Petersburg.
Trailing 2-0 from the first leg, Zenit halved the deficit after half an hour through Axel Witsel and things looked to be going their way when Basle had Marcelo Diaz sent off for two yellow cards just before half-time.
But Basel goalkeeper Yann Sommer saved a late penalty from Roman Shirokov and the hosts hit a post and missed an open goal as they threw all they had at their opponents, with a 1-0 win on the night not enough to avoid defeat.
There was more joy for Rubin Kazan as they beat Levante 2-0 on the night after a goalless first leg, albeit with the goals from Jose Rondon and Vladimir Dyadyun coming in extra time.
Rondon struck in the 99th minute for his fifth goal in the Europa League this season with substitute Dyadyun sealing victory with eight of the 30 extra minutes remaining after Levante had committed men to finding the equaliser which would have seen them through on away goals.
Papiss Cisse repeated his injury-time heroics to head Newcastle into the Europa League quarter-finals as 10-man Anzhi Makhachkala finally succumbed 1-0.
The Senegal international, who fired the Magpies to a 2-1 Barclays Premier League win over Stoke on Sunday with a 92nd-minute strike, powered home Sylvain Marveaux's inviting cross in the 94th with the final touch of an intriguing contest.
Anzhi, reduced to 10 men 10 minutes into the second half when midfielder Mehdi Carcela-Gonzalez was sent off, had gone close with two minutes of normal time remaining when Mbark Boussoufa's free-kick came back off Rob Elliot's crossbar.
Fernando Torres fired Chelsea into the last eight, but also missed a penalty as last season's Champions League winners overcame Steaua Bucharest 3-1 at Stamford Bridge.
Chelsea had to overturn a 1-0 first-leg deficit to progress and levelled the tie through Juan Mata after 33 minutes, only for Vlad Chiriches to score on the stroke of half-time when the Blues failed to clear a corner.
The away goal meant Chelsea needed to score twice to be included in tomorrow's last-eight draw and captain John Terry headed in after 58 minutes and Torres netted after 71 minutes to seal victory - before the Spaniard hit the bar with an 87th-minute penalty.
Substitute Salih Ucan was the hero as Fenerbahce did enough to squeeze through with a 1-1 home draw against Viktoria Plzen.
The Turkish side were 1-0 winners in the away leg, meaning this result - secured after Ucan opened the scoring nine minutes after his injury-enforced introduction - was enough.
Vladimar Darida brought the Czech League leaders level just after the hour and David Limbersky hit a post in pursuit of a winner that never came.
Libor Kozak scored twice in the opening 10 minutes and completed a hat-trick late on as Lazio turned on the style behind closed doors, winning 3-1 against against Stuttgart in Rome.
Armed with a 2-0 lead from the first leg in Germany, Lazio made an ideal start tonight and left Stuttgart's hopes in tatters before they even had a chance to settle.
Tamas Hajnal pulled one back for Stuttgart just after the hour mark, but they needed five. Kozak's third killed off the hopes of even the most optimistic Swabian.
The games was played behind close doors because UEFA last month ordered the Italians to lock out supporters from their next two home matches in Europe after crowd trouble during their last-32 match against Borussia Monchengladbach.
Benfica eased into the quarter-finals as they proved too strong for Bordeaux in the south of France, winning 3-2 on the night.
With the comfort of a 1-0 lead from the first leg, the Portuguese side withstood Bordeaux's early attacks before snatching a lead through Jardel's header.
Cheick Diabate gave the hosts a glimmer of hope with 16 minutes left but it was extinguished seconds later by Oscar Cardoza as Benfica made certain of the win.
The teams exchanged further goals in the dying seconds, with Jardel scoring an own goal before Cardoza replied instantly with Benfica's third, but the maths was already against the hosts.
Europa League last-16 second leg results
Rubin Kazan 2-0 Levante (after extra-time)
Zenit St. Petersburg 1-0 Basel
Inter Milan 4-1 Tottenham (after extra-time)
Bordeaux 2-3 Benfica
Chelsea 3-1 Steaua Bucuresti
Fenerbahce 1-1 Viktoria Plzen
Lazio 3-1 Stuttgart
Newcastle 1-0 Anzhi