The appointment comes 10 weeks after Spurs sacked Antonio Conte, with Cristian Stellini and then Ryan Mason holding the forth for the rest of the season.
Spurs missed out on European football for next season after finishing eighth on the Premier League table.
Postecoglou is the fourth permanent manager for the North London side since Mauricio Pochettino was sacked in 2019.
In that period, Jose Mourinho, Nuno Esperito Santo and Antonio Conte have all come and gone.
Postecoglou ends a two-year stay at Celtic, where he won two Scottish league titles and completed a clean sweep of three domestic trophies this term.
The 51-year-old is the first Australian to manage in the Premier League.
Spurs confirmed the news on Tuesday morning, tweeting: 'We are delighted to announce the appointment of Ange Postecoglou as our new head coach on a four-year contract.'
Daniel Levy, the Spurs chairman, said of the incoming head coach: "Ange brings a positive mentality and a fast, attacking style of play.
"He has a strong track record of developing players and an understanding of the importance of the link from the academy - everything that is important to our club.
"We are excited to have Ange join us as we prepare for the season ahead."
"They wanted me to extend my time at Celtic and while I am so respectful and understanding of their position, a new opportunity has been presented to me and it is one which I wanted to explore," said Postecoglou.
