Morrison, a Turnbull ally who served as treasurer, won a party vote 45-40 in Canberra on Friday.
The challenge was mounted by former home affairs minister Peter Dutton after a party backlash against Turnbull's more moderate approach and his plan to embed carbon emissions targets in law.
Dutton, who favours slashing migrant numbers and pulling Australia out of the Paris climate agreement, was the sole candidate to be prime minister until Thursday when Morrison and Foreign Minister Julie Bishop entered the fray to try to halt his power grab.
Turnbull, who had pledged to quit parliament after his near three-year reign came to an end, survived one attempt to oust him on Tuesday, but ministers then began defecting, throwing the government into crisis.
On Thursday Dutton presented Turnbull with a petition showing he had lost the support of the majority of Liberal MPs, triggering the leadership vote.
Turnbull accused Dutton of bullying and intimidation in the move to knife him.
His departure from politics will spark a by-election for his Sydney seat, threatening the government's one-seat parliamentary majority.
Turnbull had to officially inform Governor-General Peter Cosgrove of events and recommend Morrison form a new government.
The PM-elect, an evangelical Christian and former immigration minister, took credit for "stop the boats", a harsh policy to halt asylum-seekers from entering Australia, and is further to the right than Turnbull but not as hardline as some in the party.
No Australian leader has managed to serve out a full term since John Howard lost the 2007 election.