Postecoglou promises exciting times to come at Spurs
The Australian has signed a four-year deal after taking Celtic to the Scottish domestic treble this season.
The 57-year-old will officially join the London club, who finished a disappointing eighth in the Premier League, on July 1.
Antonio Conte left Spurs by "mutual agreement" in late March, with Cristian Stellini and then Ryan Mason taking temporary charge of the struggling club.
Postecoglou said he was relishing the task of turning around the club's fortunes.
"It's great to finally get started as manager of this great football club," he said in a video message posted on the club's social media accounts on Friday.
"I'm really looking forward to our task ahead. I think it's an exciting opportunity for us to now set off on a new direction, play football and create an environment that embodies the values and traditions of this fantastic football club.
"Hopefully a team that you can all be proud of, and more importantly get excited by."