Van de Beek broke the deadlock in the 15th minute when he received a through pass from Hakim Ziyech and slot it past Hugo Lloris.
Tottenham, without injured Harry Kane and suspended Son Heung-min, nearly conceded a second 10 minutes later, but Van de Beek's shot was well saved by the French goalkeeper.
Mauricio Pochettino's side suffered another blow in the 32nd minute, when defender Jan Vertonghen's face hit teammate Toby Alderweireld's head. He received treatment on the field and returned to action six minutes later but was forced to leave the pitch after only 40 seconds.
Tottenham's best chance came in the second half from Vertonghen's substitute Moussa Sissoko, whose drive from 35 minute was just wide of the right post.
Ajax, who had beaten Real Madrid and Juventus en route to the last four, almost extended their lead to 2-0 10 minutes from time, but David Neres' shot hit the post and rebounded to safety.
It was the first time that Spurs failed to score in the Champions League in 21 matches. Ajax have won a Champions League semifinal match for the first time since 1996 when they beat Panathinaikos 3-0.
The second leg will be played at Amsterdam on Wednesday May 8.