Simons, 22, who has cost Spurs a €60 million transfer fee, has signed a five-year contract with the option of a two-year extension.
"I'm really happy and can't wait to get going. I've been dreaming of this for a long time," he told the club's official website.
"It's a great club and when I met the head coach I knew straight away that this was the right place for me.
"I will bring flair to the team, but also hard work and discipline. I want to do everything I can to win, for the team and also for the fans."
Tottenham had been braced for another transfer battle for Simons, having missed out on Eberechi Eze as he joined rivals Arsenal, while Morgan Gibbs-White signed a new contract at Nottingham Forest.
Chelsea held talks earlier in the window but never launched an official bid, instead focusing on completing a £40 million move for Manchester United winger Aleandro Garnacho.
That left the door open for Spurs to pursue a deal for the former Paris Saint-Germain player, who joins Mohammed Kudus, Joao Palhinha and Kota Takai as their new arrivals this summer.
Spurs boss Thomas Frank said: "I'm really pleased we have brought Xavi in - he is a great addition to the squad.
"He is still young; however, he already has good experience and, over the past few years, has played a lot of games at the top level.
"Xavi has proven his ability to get goals and assists, both from the number 10 position and left-wing.
"He also has a great eye for his teammates in terms of unlocking defences, and I know he will come in and be part of a good team that is already working hard together."
Simons came through the Barcelona academy and had spells with PSG and PSV Eindhoven before landing at RB Leipzig.
In 78 appearances for Leipzig, he recorded 22 goals and 24 assists, while also netting five times in 28 appearances for the Netherlands.