The 21-year-old had previously agreed a six-year deal and completed his medical on Wednesday and was given the number six shirt for the London club.
British media had reported that Tottenham were expected to pay a club-record fee of some 42 million pounds ($53.76 million) for the defender, who scored six goals in 45 games for Ajax last season and was their player of the year.
"This is a massive opportunity for me because of what Tottenham represent, not just here in England but around the world," Sanchez told the club's official website (www.tottenhamhotspur.com).
"It is a huge step forward in my career and will allow me to continue developing and achieving big things. This is a great chance," he added.
The defender joined Ajax in June 2016 from Colombian club Atletico Nacional, where he made his debut aged 17, and was part of their Copa Libertadores-winning side last year.
Sanchez made his senior Colombia debut last November and has made two appearances for the national side.
Tottenham won their opening game at Newcastle United but lost their first home game, at temporary home Wembley Stadium, to Chelsea on Sunday.
(Reporting by Simon Evans; Editing by Clare Fallon)
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