He won the Tory leadership contest after rival Penny Mordaunt failed to secure enough backing from MPs.
In his first speech, Mr Sunak said bringing his party and the UK together would be his "utmost priority".
Mr Sunak will become the UK's first British Asian prime minister and the youngest for more than 200 years.
Mr Sunak - a 42-year-old practising Hindu - is expected to take office on Tuesday after being formally appointed by the King.
He replaces Liz Truss, following her resignation just 45 days into her tumultuous premiership last week.
After chairing her final cabinet meeting at 09:00 BST, the outgoing prime minister will make a statement outside No 10 before travelling to Buckingham Palace for her final audience with the King.
This will be followed by Mr Sunak's first audience with the monarch, during which he will be invited to form a government.
He will then travel to Downing Street to make a statement at about 11:35, before entering No 10.
President Joe Biden has said he plans to call Mr Sunak to "offer his congratulations" after his meeting with the King, the White House Press Secretary said.
Mr Sunak takes office as the third Conservative prime minister since the party won the last general election in 2019 - prompting Labour to lead calls for an early general election as a result.
In the speech - which took place behind closed doors - Mr Sunak told them this was going to be a tough period and ruled out an early general election, MP Simon Hoare said.
The former chancellor said the Conservative Party, trailing Labour in the polls by huge margins, was facing an "existential threat", but could win the next election by uniting.
Later, in a TV address that lasted less than two minutes, Mr Sunak promised to serve with "integrity" and thanked outgoing Prime Minister Liz Truss for leading the country during "exceptionally difficult circumstances".
"The United Kingdom is a great country but there is no doubt we face a profound economic challenge," he said. "We now need stability and unity and I will make it my utmost priority to bring our party and our country together."
Mr Sunak's coronation, in effect, as Tory leader caps a rapid political comeback by the former chancellor after he lost out to Ms Truss in the last leadership contest during the summer.
Ms Truss - whose tax-cutting agenda Mr Sunak dismissed as "fairytale economics" - tweeted to congratulate him and offer her "full support".
Her successor will take office at a time of economic crisis and pressure on public finances, exacerbated by Ms Truss's mini-budget, most of which has been scrapped.
Conservative Party Chairman Jake Berry said it was time for the party to "unite four-square behind Rishi" after a period of intense political turmoil under Ms Truss's premiership.
Mr Sunak's leadership rival Penny Mordaunt withdrew from the latest contest minutes before the result was announced, admitting it was "clear that colleagues feel we need certainty today".
"This decision is an historic one and shows, once again, the diversity and talent of our party," Ms Mordaunt said. "Rishi has my full support."
Ms Mordaunt took the leadership race to the wire, but came under pressure to unite behind Mr Sunak after former Prime Minister Boris Johnson withdrew from the contest on Sunday.
Mr Johnson - who resigned as prime minister only seven weeks ago - claimed he had enough support to stand but acknowledged now was "not the right time".
Sources told the BBC that Mr Sunak and Mr Johnson met on Saturday as Tory MPs decided who to back in the second Tory leadership contest in four months.
Mr Sunak was chancellor in Mr Johnson's government and within weeks of taking the role had to steer the UK economy through the Covid-19 pandemic.
But he quit as chancellor in July after the government was dogged by scandals, giving momentum to a cabinet mutiny that forced Mr Johnson from office.
First elected as an MP for the constituency of Richmond in North Yorkshire in 2015, Mr Sunak has experienced a meteoric rise to the apex of British politics.
Before becoming an MP, Mr Sunak worked in finance and is thought to be one of the richest MPs.
But his family's finances came under intense scrutiny earlier this year, when the tax affairs of his wife, Akshata Murty, were placed in the spotlight.
Opposition parties have been clamouring for a general election, arguing that Mr Sunak does not have a democratic mandate to become prime minister.
Mr Sunak will be the fourth consecutive prime minister - after Theresa May, Mr Johnson and Ms Truss - to take the role of leader without a general election, although Mrs May and Mr Johnson both won general elections as PMs in 2017 and 2019.
The next general election is not due to take place until January 2025 at the latest, after the Conservatives won a landslide majority in 2019.
Mr Sunak is under no obligation to call an early election under the UK's parliamentary political system.
Labour's deputy leader Angela Rayner said Mr Sunak had been crowned prime minister "without him saying a single word about how he would run the country and without anyone having the chance to vote".
Liberal Democrat leader Ed Davey said Tory MPs had "installed another out-of-touch prime minister with no plan to repair the damage and without giving the British people a say".
Scotland's First Minister Nicola Sturgeon said Mr Sunak should call an early general election and must not "unleash another round of austerity".
-- Courtesy of BBC News