The Dutch side were well on top when opening the scoring and came closest to netting the second goal of the night through David Neres's shot in the second half, which came back off the post.
Tottenham had spells on top after switching from a three-man defence to four at the back, but they managed just one shot on target across the 90 minutes and now have it all to do in next week's return fixture in Amsterdam.
It was the current Ajax crop who started the match on top, soon finding their groove and deservedly edging in front through the first serious attacking move of the match.
Tottenham's supporters were starting to grow a little agitated as Ajax created another good opportunity five minutes later, but Van de Beek was this time denied by Lloris at his near post when he had the option to return the ball to Dusan Tadic in a better position.
Pochettino changed shape following his side's slow start and that paid off as Fernando Llorente and Toby Alderweireld headed over the crossbar, either side of the home side losing Jan Vertonghen to a nasty facial injury.
Spurs, who had not led at half time in any of their previous 10 Champions League matches this season, showed a lot more attacking impetus in the second half.
Dele Alli had a volley kept out five minutes after the restart - the closest the Lilywhites would come to a goal - and the same player headed over from a good position soon after.
Unlike their opponents in the first half, however, Tottenham could not make their spell on top count as numerous attacks broke down in and around the box.
To emphasise Pochettino's lack of game-changing options on the bench, the Spurs boss could only bring on Ben Davies and Juan Foyth in like-for-like changes at full-back for the final 10 minutes of the game.
It was Ajax who came closest to scoring the second goal of the match as Neres fired the ball across Lloris and against the foot of the post following a positive attacking move.
Tottenham may have avoided conceding a late second, but they still have plenty to do in the Netherlands next week as they need to score at least once.
TOTTENHAM (3-5-2): Lloris; Sanchez, Alderweireld, Vertonghen (Sissoko 39'); Trippier (Foyth 80'), Eriksen, Wanyama, Alli, Rose (Davies 79'); Moura, Llorente
AJAX (4-2-3-1): Onana; Veltman, De Ligt, Blind, Tagliafico; Schone (Mazraoui 65'), De Jong; Ziyech (Huntelaar 87'), Van de Beek, Neres; Tadic