The Dane moves to north London on a three-year deal, replacing Ange Postecoglou in the dugout after the Australian was sacked last Friday.
Spurs paid an approximate £10 million fee to release him from his Brentford contract, which had another two years left to run.
Analysis: The reasons why Frank is not an ideal choice for Spurs
Frank's appointment was partially held up due to a conflict over the formation of his backroom staff, namely the inclusion of assistant coach Justin Cochrane.
Brentford were keen to keep the highly-rated Cochrane, but he will now accompany Frank at Spurs, returning to the club where he was previously a youth coach.
Head of athletic performance Chris Haslam and analyst Joe Newton have also left Brentford to make the switch across London.
A statement on Spurs' website read: "We are delighted to announce the appointment of Thomas Frank as our new Head Coach on a contract that runs until 2028.
"Thomas has extensive experience in English football, having joined Brentford in 2016 - since becoming one of the longest-serving current managers in the Premier League.
"During his time at Brentford, he transformed the Club, moving them up from the Championship to an established Premier League side, consistently and significantly outperforming expectations for an extended period of time.
"In Thomas, we are appointing one of the most progressive and innovative head coaches within the game. He has a proven track record in player and squad development, and we look forward to him leading the team as we prepare for the season ahead."
Frank took charge of Brentford in October 2018 and guided them into the Premier League via the Championship playoffs in 2021.
He helped the Bees establish themselves as regulars in the top flight during four seasons, despite a relatively modest budget.
Brentford ended the most recent campaign in 10th place and now takes over a Tottenham side that finished way down in 17th.
Spurs' poor domestic campaign led chairman Daniel Levy to axe Postecoglou, despite delivering success in the Europa League.
Postecoglou led Spurs to their first piece of silverware in 17 years with a victory over Manchester United in the Bilbao final, but was shown the door just 16 days later.
Bees salute Frank
Brentford director of football Phil Giles said in a statement: "From the moment he (Frank) replaced Dean Smith, he understood what we were trying to build and his wisdom, coaching ability, and emotional intelligence have helped transform the club.
"But it's not just what you see on the pitch. He forged a special connection with our fans, helped develop and improve players, and was instrumental in implementing the culture that has seen Brentford go from strength to strength.
"We will never forget Thomas, but now it is time to thank him and take the next steps in our journey with a new leader who we believe can be just as successful and influential."