The Conservative leader warned that the UK is facing a “profound economic crisis” in his first speech since being appointed by the King and before launching a reshuffle.
Boris Johnson loyalists who stayed close to Ms Truss, including Jacob Rees-Mogg, were among the 11 ministers who were out of government.
But Downing Street announced that Mr Hunt, brought into the Treasury to fix the economic crisis which began under the last PM, would stay in the role ahead of his Halloween financial statement.
Mr Hunt, who was seen as safe in the role with the new leader eager to reassure the financial markets after the turbulence of the Truss administration, warned “it is going to be tough”.
Mr Raab, one of Mr Sunak’s strongest backers who was sacked by Ms Truss, was rewarded with Deputy Prime Minister and Justice Secretary, roles he held under Mr Johnson.
Simon Hart was brought in as Chief Whip to restore party discipline, with Mr Sunak having warned the party: “Unite or die.”
Oliver Dowden, who resigned as party chairman after a disastrous double by-election loss under Mr Johnson, was appointed Chancellor of the Duchy of Lancaster in the Cabinet Office.
James Cleverly was kept on as Foreign Secretary and Ben Wallace as Defence Secretary, signalling Mr Sunak was not entirely breaking with the past two administrations.
More than an hour after Ms Truss defended her economic strategy in her farewell speech from Downing Street, Mr Sunak stood outside No 10 criticising her brief tenure.
Mr Sunak said his predecessor, whose 49 days in office made her the shortest-lasting PM in history, was “not wrong” to want to drive up growth, describing it as a “noble aim”.
“But some mistakes were made. Not born of ill will or bad intentions – quite the opposite in fact. But mistakes nonetheless,” he added.
“I’ve been elected as leader of my party and your Prime Minister in part to fix them – and that work begins immediately.”
Mr Sunak, 42, became the UK’s first Hindu PM, the first of Asian heritage and the youngest for more than 200 years when he was appointed by Charles at Buckingham Palace.
The pound soared to the highest level since before Ms Truss’ disastrous mini-budget and the cost of Government borrowing dropped.
Eleven ministers who were either in or attended Cabinet returned to the back benches after Mr Sunak began his cull of Ms Truss’ top team.
Simon Clarke and Kit Malthouse followed Mr Rees-Mogg out of the Cabinet, while Tory chairman Sir Jake Berry and chief whip Wendy Morton also departed.
Ms Truss had used her valedictory speech to stress the need to be “bold” as she defended her tax-cutting ideals despite being forced to reverse most of her policies.
She made no apologies for her mini-budget and stressed the need for lower taxes, before wishing Mr Sunak “every success, for the good of our country”.
After Mr Sunak was pictured shaking the King’s hand during a formal handover of power in which the monarch was “graciously pleased to accept” Ms Truss’s resignation, he sought to explain why he was now Prime Minister.
“Right now our country is facing a profound economic crisis,” he warned, blaming the lingering aftermath of the coronavirus pandemic and Vladimir Putin’s destabilising war in Ukraine.
He vowed to place “economic stability and confidence at the heart of this Government’s agenda”, after the financial chaos triggered by Ms Truss.
“This will mean difficult decisions to come,” he said, but he promised to repeat the “compassion” he showed during the coronavirus pandemic.
Mr Sunak vowed not to leave future generations “with a debt to settle that we were too weak to pay ourselves”.
“I will unite our country, not with words, but with action,” he said, vowing to run a Government of “integrity, professionalism and accountability at every level”.
“Trust is earned and I will earn yours,” he said in the speech lasting nearly six minutes.
He entered Downing Street after winning the swift Conservative leadership contest on Monday, with rivals Mr Johnson and Penny Mordaunt pulling their campaigns before a single vote was cast.
Mr Johnson, who had plotted a comeback less than two months after he resigned following a series of scandals, offered his “congratulations” to Mr Sunak just after his speech ended.
He welcomed the “historic day” and said “this is the moment for every Conservative to give our new PM their full and wholehearted support”.
As he became the third Prime Minister based on the mandate won by Mr Johnson in the 2019 general election, Mr Sunak vowed to deliver on that manifesto’s promises.
“All I can say is that I am not daunted. I know the high office I have accepted and I hope to live up to its demands,” he said, having ruled out a general election in talks with MPs.
Instead he pledged to “build a Government that represents the very best traditions of my party” as he prepares for a widely anticipated Cabinet reshuffle.
Mr Sunak will look to build a new top team with an eye on the task of uniting his fractious Tory party.
Long-time backers Dominic Raab, the former justice secretary, Commons Treasury Committee chairman Mel Stride and ex-chief whip Mark Harper have also been tipped for jobs.
In her brief exit speech, Ms Truss celebrated reversing the national insurance hike imposed by Mr Sunak when he was chancellor.
She warned 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.”