He becomes the first Australian to manage a team in the Premier League and will join Spurs on July 1.
"Ange brings a positive mentality and a fast, attacking style of play," Spurs chairman Daniel Levy said in a statement.
"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."
The 57-year-old ex-Australia international, who guided Celtic to a domestic treble this season, will replace Antonio Conte as Spurs' permanent boss.
They were linked with Feyenoord's Arne Slot, Bayer Leverkusen's Xabi Alonso and former Bayern Munich manager Julian Nagelsmann before landing on Postecoglou.
Following Conte's acrimonious exit in March, Cristian Stellini and Ryan Mason took interim charge at the north London side.
Italian Conte took charge of Spurs in 2021 but became the seventh manager to fail to end their 15-year trophy drought. Spurs last lifted silverware in 2008 when they won the League Cup.
They finished eighth in the league after a dismal campaign and missed European football for the first time since the 2009-10 season.
Postecoglou faces many challenges in his bid to spark a revival at Spurs, chief among which is resolving the future of talismanic striker Harry Kane.
Kane, who this year became both England and Spurs' record goalscorer, has one year left on his contract. He has been linked with a move away from the Premier League side, with both Manchester United and Real Madrid touted as possible destinations.
MOMENT OF PRIDE FOR AUSTRALIAN FOOTBALL
Postecoglou has enjoyed success pretty much wherever he has coached over the last 26 years.
He joined Celtic from Yokohama F Marinos, a J1 League club, in June 2021 on a 12-month rolling contract and helped them reclaim the Scottish Premiership title in his first season.
In Japan, he guided Yokohama to their first league title in 15 years in 2019.
His other coaching experiences include spells at Brisbane Roar and Melbourne Victory, as well as a four-year stint with Australia during which he led the Socceroos to the 2014 World Cup and the following year secured their first major title at the Asian Cup.
"This new chapter at Spurs for Ange is a testament to his enduring determination, skill, and vision as a leader," Football Australia CEO James Johnson said.
"It is also a moment of great pride for Australian football...
"(It) highlights this current golden age as we see more of our coaches making an impact on football around the globe and a steady presence of players across European leagues, particularly in Great Britain."
