The new Conservative leader was invited to form a government by Charles when they met at Buckingham Palace on Tuesday after Ms Truss departed as the shortest-serving prime minister in history.
Mr Sunak, 42, is the UK’s first Hindu PM, the first of Asian heritage and the youngest for more than 200 years.
The King and Mr Sunak were pictured shaking hands as they met in Buckingham Palace.
The former chancellor was expected to address the nation from Downing Street a short while later.
Ms Truss used her farewell speech to stress the need to be “bold” as she defended the tax-cutting agenda that sparked economic chaos and led to her downfall.
After seven chaotic weeks as prime minister, Ms Truss celebrated reversing the national insurance hike imposed by Mr Sunak when he was chancellor.
She warned from Downing Street that the nation continues to “battle through a storm”, but insisted she believes “brighter days lie ahead”.
“From my time as prime minister, I’m more convinced than ever that we need to be bold and confront the challenges we face,” she added.
“We simply cannot afford to be a low-growth country where the Government takes up an increasing share of our national wealth and where there are huge divides between different parts of our country. We need to take advantage of our Brexit freedoms to do things differently.”
Ms Truss made no apologies for the disastrous mini-budget and continued to stand by her tax-cutting ideals, despite being forced to reverse most of her policies when new Chancellor Jeremy Hunt was brought in to clear up the mess.
She cited one of Brexit’s benefits as “lower taxes, so people keep more of the money they earn”, before wishing Mr Sunak “every success, for the good of our country”.
In the speech lasting three minutes and seven seconds, Ms Truss quoted Roman philosopher Seneca to say: “It is not because things are difficult that we do not dare. It is because we do not dare that they are difficult.”
Ms Truss thanked her family and her short-lived Downing Street team during the speech in front of a relatively small crowd of supporters that included her daughters Frances and Liberty, husband Hugh O’Leary and Deputy Prime Minister Therese Coffey.
“We continue to battle through a storm but I believe in Britain, I believe in the British people and I know that brighter days lie ahead,” she ended her speech by saying.
The King was “graciously pleased to accept” her resignation, Buckingham Palace said.
Mr Sunak will look to build a new Cabinet that might unite a fractious Tory party.
He won the Tory leadership contest on Monday without a vote after rivals Penny Mordaunt and Boris Johnson dropped out of the race.
Allies hope he will stabilise the party following Mr Johnson’s dramatic downfall and his successor’s fleeting but tumultuous tenure.
Mr Sunak is expected to quickly begin assembling a top team to portray a measure of stability to both the Conservatives and the country.
Long-time backers Dominic Raab, the former justice secretary, Commons Treasury Committee chairman Mel Stride and ex-chief whip Mark Harper have been tipped for jobs.
While not confirmed, Mr Hunt, who replaced Kwasi Kwarteng as Chancellor amid economic turbulence, is expected to remain at the top of the Treasury.
Mr Hunt has been working towards a highly-anticipated Halloween statement on the Government’s medium-term fiscal plans, complete with independent forecasts.
Mr Sunak has ruled out allowing the early general election demanded by opposition parties as the Tories move on to their third prime minister on the mandate won by Mr Johnson in 2019.
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