Andre Villas-Boas' side crashed out through the lottery of spot kicks despite fighting back to secure a draw against the impressive Swiss giants
The match had finished level at 2-2 in normal time, with Clint Dempsey's early strike putting Spurs in control of the tie, but they were quickly pegged back thanks to the influential Mohamed Salah's near-post strike.
Basel capitalized on slack defending to establish a stranglehold on the tie as Aleksandar Dragovic rifled home from close range after Brad Friedel had parried a Fabian Schar header.
Plucky Tottenham was not to be denied though, as Dempsey netted his second following a searching pass from substitute Tom Huddlestone but Jan Vertonghen's red card prior to the final whistle left the visitors with an uphill task for extra-time.
They held on with last-gasp defending against Murat Yakin's outfit but ultimately to no avail as Huddlestone and Emmanuel Adebayor missed from 12 yards to seal the North London club's fate.
Andre Villas-Boas made four changes - three enforced - to the side that drew 2-2 with the Swiss outfit in the first leg as injured trio Gareth Bale, Aaron Lennon and William Gallas were replaced by Michael Dawson, Gylfi Sigurdsson and Dempsey in the starting XI while Walker also came in for the benched Benoit Assou-Ekotto.
The hosts made just two changes meanwhile, with Park Joo-Ho and Philipp Degen taking their places at the expense of Kay Voser and Markus Steinhofer.
In a rather dull opening, Schar was the first to chance his arm, firing in a low free kick which flew just wide of Friedel's near post.
Schar and his defensive partner Dragovic had been largely composed in the opening exchanges, with both rearguards on top, but the latter was badly at fault as Tottenham seized a crucial advantage in the tie at the halfway point of the first half.
Vertonghen's speculative pass through the middle look pretty harmless but Dragovic failed to deal with it and in nicked Dempsey, who rounded the keeper before coolly rolling the ball into the unguarded net.
The lead, however, did not last long.
Dembele was caught napping in midfield by Streller, who laid the ball off to the onrushing Salah, who - despite a heavy touch - toe-poked the ball beyond the reach of Friedel at his near post.
The Egyptian almost stole in for a second moments later but could not quite prod past the alert Friedel before fellow goalscorer Dempsey rose well at a corner only to head straight at Sommer.
Yann Sommer had to go one better before the break as he sprung to his right to push away Dembele's curling strike before getting down low to hold on to an Adebayor effort from the subsequent attack in the last meaningful action of the half.
The Togolese striker registered the first notable attempt after the interval as he nodded down a corner which Sommer was able to gather.
From a set-piece of its own though, Basel moved ahead for the first time on the night and strengthened its grip on the tie.
A flick-on by Schar at the near post from Stocker's delivery was inexplicably pushed out by Friedel to the unmarked Dragovic who atoned for his earlier error by slamming into the roof of the net from a couple of yards out.
The Swiss defender almost doubled his tally for the evening as he connected perfectly with Fabian Frei's inswinging free kick, but his header floated just wide of the upright, with Friedel rooted to the spot.
Spurs were becoming increasingly anxious to find the equalizer they craved but were unable to break through a resolute Basel defense, with the hosts always looking a threat on the counter, particularly through the exciting Salah.
Yakin's side pressed its opponents effectively too, and it almost paid off as Marcelo Diaz capitalized on a poor Lewis Holtby pass to proceed goalwards and fire a stinging effort at Friedel, who dealt well with the strike.
Given the defiant nature of the Swiss side's defence, Spurs found their deserved but slightly unexpected equalizer completely out of the blue.
Substitute Huddlestone sent a raking pass in behind Joo-Ho to Dempsey, who managed to send a volley goalwards. The ball somehow crept underneath Sommer, who appeared to dive too early, to breathe new life into the tie.
Joo-Ho, who was enduring a nightmare few minutes, nearly turned the ball into his own net from a Kyle Walker cut-back just moments after Friedel had denied Basel again from Streller's diving header in what was becoming an increasingly end-to-end affair.
Vertonghen struggled with the constant movement of a willing Streller all evening and as the tie approached extra-time, Spurs task was made all the more difficult after the Belgian tripped the striker as he rushed through on goal, and as the last man, referee Olegario Benquerenca had no choice but to dismiss the center back.
There was to be no seperating the two sides after 90 minutes but it was the Swiss side wich used its numerical advantage in the opening period of extra-time, with Mohamed El-Nenny crashing a strike against the upright as the Egyptian starlet came agonising close to restoring his side's lead.
Spurs were beginning to tire in midfield, while their opponents sensed their chance and came close once more as Diaz side-stepped a weak challenge and curled just wide, with Friedel sprawling.
The veteran striker Alexander Frei emerged from the bench to replace Salah for the final minutes and engineered an opportunity within seconds of his arrival.
Streller's neat layoff found the substitute in space on the left-hand side of the area, but having adjusted his footing, he could not find a finish as Dawson blocked heroically.
Basel huffed and puffed for their winner but it was not to be as the tie moved onto a dramatic shootout.
Huddlestone's effort was saved by Sommer, before Adebayor blazed horribly over. It left Chilean Diaz with the task of putting his side into the semifinals, which he did by duly dispatching the spot kick.