The 34-year-old Portuguese, who was sacked by Chelsea after a 257-day reign at Stamford Bridge, returns to the game with a three-year contract and within four months of his dismissal.
"Tottenham is a great club with a strong tradition and great support both at home and throughout the world. I feel privileged to be its coach. For me, this is one of the most exciting coaching positions in the Premier League. This is a squad any coach would love to work with. Together I believe we can bring success in the seasons ahead," said Villas-Boas.
After Redknapp's dismissal, former France coach Laurent Blanc, Wigan's Roberto Martinez, Fabio Capello and former Tottenham striker Jurgen Klinsmann were rumoured to be in the running, but the former-Porto boss emerged as the preferred candidate.
"He has an outstanding reputation for his technical knowledge of the game and for creating well-organised teams capable of playing football in an attractive and attacking style," Spurs chairman Daniel Levy told the club's official website.
"Andre shares our long-term ambitions and ethos of developing players and nurturing young talent, and he will be able to do so now at a new world-class training centre."
Villas-Boas will bring fitness coach Jose Mario Rocha and Daniel Sousa, head of opposition scouting. Both Rocha and Sousa were with him at Chelsea.
"It is important we now look to develop the potential within the squads at all levels, while strengthening the first team in the summer in key positions with players who will become part of the future success of the club," said Levy.
Tottenham finished fourth in the Premier League last season. They kick-off this season away to Newcastle, August 18. Villas-Boas first encounter with Chelsea will be October 20 at White Hart Lane.