The endorsement gives the former chancellor a boost ahead of the second ballot on Thursday, which will eliminate the least popular candidate.
Mr Hunt, who along with Chancellor Nadhim Zahawi failed to garner the 30 votes needed to progress to the next stage of the contest, described Mr Sunak as “one of the most decent, straight people with the highest standards of integrity” in politics.
Mr Sunak, whose resignation from No 11 helped trigger the Tory leadership race, topped Wednesday’s ballot, as trade minister Penny Mordaunt emerged as his leading rival in second place.
The former defence secretary surged ahead of Foreign Secretary Liz Truss, despite being seen as a relative outlier before the race began.
Former equalities minister Kemi Badenoch, senior backbencher Tom Tugendhat and Attorney General Suella Braverman are also among the final six candidates.
Ms Braverman appears most under threat in the second ballot after squeaking through to the round by two votes.
Ms Truss is seeking to unite the right of the party, as subsequent voting from Thursday will whittle down the field until two candidates are left.
“Now is the time for colleagues to unite behind the candidate who will cut taxes, deliver the real economic change we need from day one and ensure Putin loses in Ukraine,” a spokeswoman for the Foreign Secretary said.
Ms Truss is set to highlight her experience in the Cabinet and Treasury at a campaign launch event on Thursday, in an apparent bid to portray herself as more competent on the economy than Ms Mordaunt.
Mr Hunt, who has held the offices of health and foreign secretary, said he is backing Mr Sunak because of his character and approach to the economy.
He said in a statement: “I’ve given this a lot of thought and I’m supporting Rishi Sunak. And that’s for two reasons.
“First of all, the big challenge we face now is economic, and this is someone of formidable ability, who has been thinking about the right thing to do for our economy, for families up and down the country, very hard over the last two years.
“But in the end it’s not about policy. I’ve been around long enough to know that politics is really about character and Rishi is one of the most decent, straight people with the highest standards of integrity that I have ever met in British politics and that’s why I would be proud to have him as my next Prime Minister.”
Mr Hunt had earlier condemned “smears and attacks” by rival camps, after Johnson loyalists accused Mr Sunak’s team of using “dark arts” following claims they tried to “syphon off” votes to ensure Mr Hunt cleared the threshold to enter the contest because they believed Mr Sunak would beat him in a run-off vote of party members.
Sir Graham Brady, the chairman of the Conservative 1922 Committee overseeing the contest, read out the results of Wednesday’s vote in a crowded Committee Room 14 in the House of Commons.
Mr Sunak was on 88, Ms Mordaunt on 67, Ms Truss, 50, Ms Badenoch, 40, Mr Tugendhat, 37, and Ms Braverman scraped through on 32.
Mr Zahawi, brought in by Mr Johnson after Mr Sunak’s resignation, got 25 and Mr Hunt only 18.
The remaining candidates attended a hustings organised by the 1922 Committee of backbenchers in the evening, with loud banging greeting them from inside the Commons committee room as they arrived.
Zahawi-backer Jonathan Gullis suggested the campaigns should now get behind a single standard-bearer for the party’s right-wing.
But, in conceding defeat, Mr Zahawi declined to announce his backing of a favoured candidate, saying he does not “intend to make any further intervention”.
The first round of voting came after Downing Street was forced to deny running a “stop Sunak” smear campaign as the battle grew increasingly bitter.
The caretaker Prime Minister’s press secretary insisted that Mr Johnson is “staying neutral” despite his remaining arch-loyalists throwing their support behind Ms Truss.
Culture Secretary Nadine Dorries accused the former chancellor’s campaign of deploying “dirty tricks” to benefit his campaign and backed Ms Truss as the Brexiteers’ candidate.
Brexit Opportunities Minister Jacob Rees-Mogg accused Mr Sunak of having implemented “economically damaging” policies.
Mr Johnson’s press secretary declined to say whether Downing Street remains supportive of Mr Sunak, whose resignation helped end Mr Johnson’s grip on No 10.
Transport Secretary Grant Shapps, who abandoned his own leadership bid to back Mr Sunak, denied claims from Ms Dorries that the campaign has engaged in dirty tricks.
“Simply, in this case it just didn’t happen,” he said.
Ms Mordaunt beat Ms Truss into third place after officially launching her campaign by telling colleagues who had been fearful of losing their seats under Mr Johnson’s leadership that she is their “best shot” at winning the next election.
“I’m the candidate that Labour fear the most – and they’re right to,” she told Conservatives at Westminster’s Cinnamon Club.
Tory MPs will continue to vote in subsequent rounds until two candidates are left, who will then battle it out over the summer to win the support of Conservative members, with their choice of the next prime minister being unveiled on September 5.
Mr Johnson will formally tender his resignation to the Queen to make way for his successor the following day, his official spokesman confirmed.