Donny van de Beek's well-worked goal secured Ajax a narrow advantage for next week's second leg.
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It is advantage Ajax in this UEFA Champions League semi-final after Donny van de Beek's early away goal earned a slender lead to take into next Wednesday's second leg.
The Dutch giants' slick passing rattled Tottenham from the off and Van de Beek soon stamped their authority on the tie, finishing calmly to round off a lovely flowing move.
Spurs' hopes suffered a further blow when Jan Vertonghen was forced off with a head injury before half-time, although replacement Moussa Sissoko did give the hosts more balance.
Yet Tottenham never really carved out a meaningful opportunity and were grateful to see David Neres's effort come flush off the post in the closing stages.
View from the stadium: Daniel Thacker
The electric pre-match atmosphere underscored the enormity of this fixture for Tottenham: the moment they had waited 57 years for had arrived within six games at their new home and they were going to enjoy it. Yet if there is an opponent the occasion will not cow then it is Ajax, whose pre-match billing as fearless rang true in a first-half showing full of endeavour.
Tottenham's performance after the interval will serve as a foundation for next week's Amsterdam return, but Mauricio Pochettino's men will need as nerveless a display in the Netherlands as Ajax produced here to stand any chance of progress.
Man of the match: Donny van de Beek (Ajax)
The 22-year-old was imperious in the first half. Probing, linking, ball-winning, closing down opponents, instigating moves: he was the complete midfielder. His calmness when finding himself one-on-one with Hugo Lloris was remarkable. Although his contribution was less apparent after the break, the work-rate didn't drop. The best compliment you can pay him is that he always seemed to be in the right place at the right time.
9: Ajax are still to lose a European away game in 2018/19 (W5 D4) and have won the last four.
2: Van de Beek has scored in successive UEFA Champions League matches – away to Juventus and Tottenham.
4: Spurs' starting XI boasted four former Ajax players: Toby Alderweireld (2008–13), Vertonghen (2006–12), Davinson Sánchez (2016–17) and Christian Eriksen (2009–13).
19: At 19 years and 261 days old, Ajax defender Matthijs de Ligt is the youngest player to captain a side in the UEFA Champions League semi-finals.
3: Tottenham suffered only their third defeat in 11 UEFA Champions League home games – the other eight were wins.
21: Ajax are the first team in 21 matches to keep a clean sheet away to Spurs in the UEFA Champions League.
7: Tottenham are the seventh English club to contest a semi-final in the UEFA Champions League era, after Manchester United, Liverpool, Chelsea, Manchester City, Arsenal and Leeds.
1: Ajax are the first side in European Cup history to get to the semis having come through three qualifying rounds.
57: This was Tottenham's first European Cup semi-final in 57 years, since losing to eventual winners Benfica in 1962.
What does it mean?
Tottenham have it all to do in Amsterdam next Wednesday but crucially they will have Heung-Min Son back from his one-game ban. Ajax's thrilling victories over Real Madrid and Juventus were founded on impressive away performances, and so it may prove again – albeit at home they are seven without a win in UEFA Champions League knockout phase matches (D4 L3).
Spurs continue their quest for a Premier League top-four berth with a trip to Bournemouth on Saturday. They conclude their domestic campaign at home to Everton on Sunday 12 May.
Sandwiched between these two legs for Ajax is the Dutch Cup final against Willem II on Sunday. They finish their Eredivisie title hunt at home to Utrecht on 12 May and away to De Graafschap three days later.