The palace said he would be sworn in at 5pm (2000 AEDT) on Thursday.
Anwar's appointment caps a three-decade long political journey from a protege of veteran leader Mahathir Mohamad to a prisoner convicted of sodomy to opposition leader and, finally, prime minister.
A general election on Saturday ended in an unprecedented hung parliament with neither of two main alliances, one led by Anwar and the other ex-premier Muhyiddin Yassin, immediately able to secure enough seats in parliament to form a government.
The 75-year-old Anwar has time and again been denied the premiership despite getting within striking distance over the years: he was deputy prime minister in the 1990s and the official prime minister-in-waiting in 2018.
In between, he spent nearly a decade in jail for sodomy and corruption in what he says were politically motivated charges aimed at ending his career.
The uncertainty over the election threatened to prolong political instability in the Southeast Asian country, which has had three prime ministers in as many years, and risks delaying policy decisions needed to foster economic recovery.
Anwar leads a multi-ethnic coalition of parties with progressive leanings while Muhyiddin's alliance reflects more conservative, ethnic Malay, Muslim views.
Anwar's coalition, known as Pakatan Harapan, won the most seats in Saturday's vote with 82, while Muhyiddin's Perikatan Nasional bloc won 73. They need 112 - a simple majority - to form a government.
The long-ruling Barisan bloc won only 30 seats - the worst electoral performance for a coalition that had dominated politics since independence in 1957.
Barisan said on Thursday it would not support a government led by Muhyiddin, though it did not make any reference to Anwar.
Muhyiddin's bloc includes the Islamist party PAS, whose electoral gains raised concern in a country with significant ethnic Chinese and ethnic Indian minorities, most of whom follow other faiths.
The decision on the prime minister came down to King Al-Sultan Abdullah Sultan Ahmad Shah, after Anwar and Muhyiddin missed his Tuesday afternoon deadline to put together an alliance.
The constitutional monarch plays a largely ceremonial role but can appoint a premier he believes will command a majority in parliament.
As premier, Anwar will have to address soaring inflation and slowing growth, while calming the ethnic tensions.
Anwar will also have to negotiate agreements with MPs from other blocs to ensure he can retains majority support in parliament.
27°C