Clint Dempsey scored an 82nd minute equaliser for Tottenham to send the quarter-final second leg into extra time, which Tottenham survived with 10 men as the match ended 2-2 and 4-4 on aggregate.
But it was all in vain as Togolese Adebayor casually leant back and fired his effort high and wide after team-mate Tom Huddlestone had seen a previous penalty saved by Yann Sommer.
Defender Aleksandar Dragovic was hero and villain for the Swiss champions, gifting Tottenham a goal with a blunder in the first half but then scoring with a header while Dempsey scored both goals for the English Premier League side, who were missing the injured Gareth Bale.
Tottenham, the only former winners of the competition in the last eight, went ahead in the 23rd minute following a dreadful mix up in the Basel defence.
Scott Parker played a pass through the middle and Dragovic should have easily cut it out but instead miscued, allowing Dempsey to nip in, round Sommer, and tuck the ball into the far corner with a cool finish.
Basel replied four minutes later when Mohamed Salah burst clear on the right and scored a neat left foot finish, after which the Swiss side took control.
Austrian Dragovic made up for his blunder four minutes after the break, snapping up a rebound to score after Fabian Schaer's header was parried at point-blank range by Brad Friedel.
Tottenham's challenge appeared to have fizzled out until Dempsey scored from close range with eight minutes to go.
The game took another twist when Tottenham had defender Jan Vertonghen sent off for tripping Marco Streller when he had a run at goal, followed by a scuffle as Basel were awarded a free kick on the edge of the area shortly afterwards.
Tottenham had their backs to the wall in extra time as Mohamed Elneny hit the post from 30 metres but survived, only to be let down by their efforts from the penalty spot.
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