Lucas Moura's second-half hat-trick, including an added-time winner, sent Spurs into their first European Cup final.
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Tottenham staged one of the most remarkable comebacks in European Cup history to reach their first final, recovering from a first-leg defeat and a 2-0 second-leg deficit to progress at the expense of Ajax.
Ajax extended their aggregate advantage as early as the fifth minute when captain Matthijs de Ligt lost Kieran Trippier to head in a corner, and though Heung-Min Son swiftly hit the post at the other end, that was a rare glimmer of hope for Spurs.
Ten minutes before half-time it was 2-0, Donny van de Beek and Dušan Tadić combining to set up Hakim Ziyech for a fine first-time finish. Spurs refused to be downhearted, and after André Onana had superbly denied Dele Alli, Lucas Moura raced through to revive their hopes. Four minutes later, the Brazilian fired in the loose ball after Onana had made another excellent stop, this from Fernando Llorente, to level on the night.
With both sides committed to attack, chances continued to flow at both ends, Ziyech's 20-metre shot coming back off the post while Jan Vertonghen struck the crossbar in a frantic finale. Just as it looked as if Ajax would resist, however, Moura swept into the net in the final seconds of added time, completing his hat-trick and a comeback for the ages.
View from the stadium: Derek Brookman
Another spellbinding UEFA Champions League match in the Johan Cruijff ArenA. Spurs controlled long spells of the first half yet went in two down at the break. After the restart, the Premier League side started stronger and deservedly levelled the match, only for Ajax to reassert themselves, but just as it looked as if their young side would withstand the late siege, Moura dashed their dreams at the last.
Man of the match: Lucas Moura
Struggled to make an impact in the first half but was centre stage in the second, his two goals in the space of 204 seconds bringing Spurs right back into it. Both were assured finishes and the Brazilian was a constant threat thereafter, holding his nerve again in the sixth minute of added time to apply the crucial finishing touch.
40: Spurs are the 40th club to reach the European Cup final.
1: Tottenham are the first team to reach their first European Cup final since Chelsea in 2008.
5: Spurs are the fifth English club to reach the final in the UEFA Champions League era, after Arsenal, Chelsea, Liverpool and Manchester United, setting a new competition record.
2: This is only the second time in 18 ties that a team has won a European Cup semi-final having lost the home first leg, after Ajax's 1996 win against Panathinaikos.
5: Lucas Moura is the fifth player to score a hat-trick in a Champions League semi-final, after Alessandro Del Piero, Ivica Olić, Robert Lewandowski and Cristiano Ronaldo.
50: This is the 50th UEFA Champions League match in which Ajax have scored the first goal, and their first defeat after doing so (W41 D8).
What does it mean?
Tottenham will play Liverpool in the final in Madrid on 1 June, the second all-English UEFA Champions League final after Manchester United's 2008 win against Chelsea on penalties in Moscow.
The second "final" (in the words of Pochettino) of Tottenham's week comes at home to Everton on Sunday. Spurs need a point to guarantee UEFA Champions League football next season. Fail and they will, of course, have a second chance.
There's no respite for Ajax. Having lifted the Dutch Cup on Sunday, attention now switches to the Eredivisie. The Amsterdam side are level on points with PSV Eindhoven but with a clear advantage in terms of goal difference, so wins at home to Utrecht (Sunday) and away to De Graafschap (Wednesday) should be enough.