Spurs exited the competition on spot-kicks in Basel on Thursday night after the quarter-final tie ended 4-4 on aggregate after extra-time.
Yann Sommer saved from Tom Huddlestone and Gylfi Sigurdsson then scored, but any hope 10-man Tottenham had of advancing to the semi-finals was blown by Adebayor's poor attempt from 12 yards.
The Togolese took a strange, slow, stuttering run up and then cleared Sommer's goal by a foot, before Marco Diaz went on to secure an easy 4-1 shootout win for Basle.
Adebayor's miss drew derision from all quarters, with fans either side of the North London divide and former players like Gary Lineker criticising the 29-year-old's spot-kick.
The striker, who scored four penalties for Spurs last season, defended himself, blaming his missed attempt partly on fatigue and the boggy sodden pitch in North-West Switzerland.
"It's tough," he said. "You run around for 120 minutes, you give everything you have in your stomach, you are tired, you play with 10 men for 30 minutes and at the end of the day you have to take a penalty.
"The pressure is there when you take the ball. It's not the reason why we missed. We had to score, but it's not easy for us.
"I have taken penalties for Arsenal and Real Madrid. It's a gamble.
"You miss or you score. I missed against Basle. I am very disappointed, but that's football. You just have to keep focused and keep going."
Tottenham's squad will now have a four-day break before reconvening ahead of their vital Premier League clash against Manchester City a week on Sunday.
With six season-defining games to come shortly, Adebayor knows he and his team-mates must forget about their Europa League disappointment fast.
"I'm very disappointed," he said. "Now we've got time off. We'll go home, see our families and forget about what happened and think about the next game."