The only goal of the first leg in North London was scored by Donny van de Beek, leaving the Dutch hosts well placed to reach the final for the first time in 23 years.
Deserved winners in last week's first leg, even if they did drop off slightly in the second half, Ajax head into the return fixture with Spurs boasting their first piece of silverware of the campaign.
The Amsterdam outfit ended their decade-long wait for success in the Dutch Cup thanks to a straightforward 4-0 win over Willem II at De Kuip on Sunday, courtesy of a Klaas-Jan Huntelaar brace and a goal apiece for Daley Blind and Rasmus Nissen.
With the first leg of a possible treble complete - they lead the way at the top of Eredivisie on goal difference with two games to go - Ajax now turn focus back to the Champions League.
This is already the furthest they have made it in the competition since 1997, knocking out three-in-a-row champions Real Madrid and many people's favourites Juventus along the way.
Ajax lost the home leg of the Madrid tie in the last 16 and drew against Juventus at the Johan Cruyff ArenA in the quarter-finals, only to recover away from home in the second legs. The roles are reversed now, of course, as they have a one-goal lead to protect.
In fact, the Dutch giants are winless in their last three Champions League home games and could become the fourth team to reach the Champions League final with more away wins than home wins, after Manchester United in 2011, Juventus in 2017 and Madrid last season.
Erik ten Hag's men have already played 17 games in this season's Champions League, having also had to come through three qualifying rounds. They have only lost one of those 17 games - their 2-1 home defeat against Madrid in the round of 16.
The size of Tottenham's task is pretty clear, then, especially when considering that Ajax are the only remaining side to have scored in every game in the competition this season. With six wins in a row under their belt, Ten Hag's young squad look well positioned to keep their incredible run going.
Recent form in Champions League: DLWDWW
Recent form (all competitions): WWWWWW
Only one of the 17 previous teams to lose the first leg at home in a Champions League or European Cup semi-final tie has progressed to the final. That side? Ajax in 1995-96, when losing 1-0 at home against Panathinaikos before winning 3-0 away.
The onus is on Tottenham's men to find the back of the net in Amsterdam. Having registered just one shot on target in the opening 90 minutes - their joint-lowest total in a Champions League game - far more will be required from Mauricio Pochettino's men on Wednesday.
Pochettino will have Son Heung-min back in his ranks this week - a huge boost for Spurs, who had to make do without their two most threatening attackers in the first leg, with injured striker Harry Kane also out of both matches.
Son, suspended last week, has scored 50% of Tottenham's goals in this season's Champions League knockout stages (four out of eight). The South Korean forward is also the highest Asian goalscorer in the competition's history (12 goals) and will provide the Lilywhites with an entirely different threat in behind, something they seriously lacked last time out.
Even with Son in their ranks against Bournemouth at the weekend, however, it was a familiar story for Spurs - who had Son and Juan Foyth sent off - as they suffered a third successive 1-0 loss, and their fourth in five matches.
Accounting for the 4-3 defeat to Manchester City in the quarter-final second leg, which would have felt like more of a victory as it took Spurs through to the semis on away goals, Pochettino's side have now lost five of their last six games.
Tottenham's slim squad has started to show as tiredness has become more and more apparent with each passing game.
However, with a top-four spot in the Premier League effectively assured of thanks to disappointing draws for Manchester United and Arsenal at the weekend, Pochettino's men must now produce one final 90-minute performance that could potentially define their careers.
Reaching the semi-finals of the Champions League, having not spent a single penny on new players in 16 months, is an incredible achievement in itself. Pochettino will want more, though, and the only way he will get his wish is if his side produce a magical performance in Amsterdam.
Recent form in Champions League: DWWWLL
Recent form (all competitions): LLWLLL
Despite his red card at the Vitality Stadium on Saturday, Son is eligible to feature in this second leg. The South Korean is certain to start, with Dele Alli and Christian Eriksen expected to join him in attack.
Jan Vertonghen was allowed to play despite suffering a horrific clash of heads with teammate Toby Alderweireld in the first leg, but he has showed no further ill effects and could return to the side after sitting out the Bournemouth match.
As for the hosts, Carel Eiting remains their only injury absentee, giving Ten Hag a full selection of regulars to choose from.
Hakim Ziyech has scored or assisted a goal in four of Ajax's five Champions League knockout games this season, including an assist for Van de Beek's goal at the Tottenham Hotspur Stadium.
Ajax possible starting lineup:
Onana; Veltman, De Ligt, Blind, Tagliafico; Schone, De Jong, Van de Beek; Ziyech, Tadic, Neres
Tottenham possible starting lineup:
Lloris; Alderweireld, Sanchez, Vertonghen; Trippier, Sissoko, Wanyama, Rose; Eriksen, Alli; Son
Head To Head
The two sides have met in only one previous European tie - Spurs winning 6-1 on aggregate in the 1981-82 European Cup Winners' Cup first round.
Tottenham's two previous Champions League away games against Dutch opposition ended in draws, with a grand total of 10 goals being scored at an average of five goals per game.
We say: Ajax 2-1 Tottenham (3-1 on aggregate)
Since the start of 2018, Spurs have won only one of their six Champions League away games - a 1-0 victory at Borussia Dortmund in this season's round of 16. Ajax may be winless in their last three home games in the competition, but they have the advantage of an away goal and are much fancied to complete the job in front of their fanatic supporters.