Mr Morrison is now the Prime Minister designate of Australia and leader of the centre-right Liberal Party.
Mr Turnbull resigned and called a Liberal Party leadership election after a majority of his MPs signed a petition calling on him to quit.
The leadership election was held immediately, with Treasurer Scott Morrison MP defeating Foreign Minister Julie Bishop MP and Mr Dutton.
Ms Bishop was eliminated in the first round after receiving the lowest number of votes, with Mr Morrison defeating Mr Dutton in the final round by 45 votes to 40.
Deputy Prime Minister Steven Ciobo has also been replaced by Josh Frydenberg as Deputy Prime Minister.
On Wednesday Mr Turnbull said, having survived one leadership challenge, he would quit if a majority of his MPs sign a petition backing a rival.
The petition calling on the Prime Minister to resign, which contained 43 signatures, was delivered by Mr Dutton to the Prime Ministers office.
Mr Turnbull defeated Mr Dutton in an initial leadership challenge on Friday, by 48 votes to 35.
Afterwards he claimed: “What we have witnessed at the moment is a very deliberate effort to pull the Liberal Party further to the right.
“A minority has by a process of intimidation persuaded people that the only way to stop the insurgency is to give in.”
Mr Morrison, also Treasurer of Australia, is considered close to Mr Turnbull, and gave the former PM his full backing during the leadership challenges.
Thus his victory will be seen as a defeat for the right-wing of the Liberal Party, led by Peter Dutton MP and former Prime Minister Tony Abbott, who had been hoping to replace Mr Turnbull with someone more conservative.
Mr Dutton is known for his hardline positions on immigration and security.
However public support for him becoming Prime Minister was low, at between 3 and 11 percent.
Mr Turnbull struggled to connect with the more conservative elements of his party and its support base.
In response to pressure from within his party earlier this week he abandoned a plan to slash carbon emissions by 26 percent, from 2005 levels, by 2030.
No Australian Prime Minister has served a full term since John Howard who served from 1996 to 2007.
Four Australian Prime Ministers since then, Kevin Rudd, Julia Gillard, Tony Abbott and now Mr Turnbull have been deposed by their own parties.
City of Sydney Councillor Christine Forster, sister of former Prime Minister Mr Abbott, is being encouraged to run for Mr Turnbull’s Wentworth seat should he resign according to Sky News Australia.
Mr Morrison will need the continued support of the National Party of Australia and Liberal Party of Queensland to govern, which he is expected to receive.