Republic of Ireland striker Troy Parrott missed a penalty as Spurs went down to Norwich in an FA Cup fifth-round shootout but his U-21 international team-mate Adam Idah converted from the spot for the visitors.
Parrott's penalty was one of two saved by Canaries keeper Tim Krul as the Premier League's bottom side set up a home quarter-final clash with Derby or Manchester United in the quarter-finals.
Tottenham had taken the lead in the 13th minute through Jan Vertonghen's thumping header but Norwich were the better side for large parts and deserved their 79th-minute equaliser, bundled home by Josip Drmic when Spurs keeper Michel Vorm spilled a shot from Kenny McClean.
Idah had come on as a substitute shortly before the leveller and Parrott replaced Alli five minutes into extra-time, to large cheers from the Tottenham faithful.
Both youngsters were only peripherally involved until the drama of penalties.
Spurs took the early advantage when Eric Dier converted and Vorm saved from McClean but Idah buried his effort in the top corner after Erik Lamela hit the bar.
Giovanni Lo Celso and Marco Stiepermann made no mistake before Parrott shot too close to Krul, who had inched off his line, and Todd Cantwell blasted in off the bar.
Gedson had to score to keep Spurs in it but his tame effort was easily stopped by Krul as Norwich won the shootout 3-2.
"That's what it's all about, that's why we do all the hard work," said Krul.
"As a little boy you want to save penalties to get your tem through to the next round."
With their Premier League top-four hopes in trouble and Champions League chances looking grim, Spurs' last realistic opportunity of silverware is over.
For Norwich, it is a first FA Cup quarter-final since 1992 and a welcome distraction from their relegation fight.
Spurs had got off to the perfect start when Vertonghen headed home Giovani Lo Celso's delivery.
But Tottenham's early promise faded as Norwich showed again why they are not being written off as relegation certainties.
Emiliano Buendia and Todd Cantwell were a constant menace with Buendia forcing a fine save from Vorm, who was playing his first game for Tottenham for 17 months.
Vorm made a dreadful mess of Kenny McLean's shot in the 78th minute, spilling the ball into the path of Drmic who bundled in the equaliser.
Tottenham finished extra time on top with the excellent Lo Celso denied by Krul and Gedson Fernandes blazing over before Krul was triumphant in the penalty shootout.