The Dane moves to north London on a deal until 2028, replacing Ange Postecoglou in the dugout after the Australian was sacked last Friday.
Spurs paid a £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 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 victory in the Bilbao finale, but was shown the door just 16 days later.