Sunak (42), whose resignation as finance minister last week helped precipitate Johnson's fall, secured support from 88 of the party's 358 MPs, with junior trade minister Penny Mordaunt (49) second with 67 votes and foreign minister Liz Truss (46) third with 50.
Nadhim Zahawi, who took over as finance minister from Sunak last week, and former foreign minister Jeremy Hunt were knocked out after failing to get the required minimum of 30 votes. They join three other contenders who dropped out the day before.
Those remaining - which also includes former equalities minister Kemi Badenoch, Attorney General Suella Braverman, Tom Tugendhat, chair of parliament's Foreign Affairs Committee - will go through to a second round on Thursday.
Subsequent ballots will be held among the Conservative lawmakers, eliminating the candidate with the fewest votes each time, to whittle the field down to a final two by July 21. The new leader will then be chosen from those two by the 200,000 Conservative party members in the country at large, and be announced on September 5.
While Sunak might be the most popular contender with his colleagues, a YouGov poll of nearly 900 party members found Mordaunt was the favourite, beating any of the others in a run-off. She had a huge lead over Sunak, who fared badly against almost all his rivals, and is now the bookmakers' favourite.
RAFT OF PROBLEMS
Whoever wins will face a daunting in-tray while having to rebuild public trust bruised by a series of scandals involving Johnson, from the breaking of Covid-19 lockdown rules to appointing a lawmaker to government despite having been told of allegations of sexual misconduct.
Britain's economy is facing rocketing inflation, high debt, and low growth, leaving people grappling with the tightest squeeze on their finances in decades. All this is set against the backdrop of an energy crunch exacerbated by the war in Ukraine, which has sent fuel prices soaring.
As the contest intensifies it has also become increasingly fractious as rival camps trade barbs and some offer a series of eye-catching tax cutting pledges.
Sunak said it was not credible to offer more spending and lower taxes, saying he was offering honesty "not fairytales".
Zahawi said he had been smeared over his personal finances while culture minister Nadine Dorries, who was fiercely loyal to Johnson and is now backing Truss, has accused Sunak's team of "dirty tricks" as part of a "Stop Liz" strategy.
"I believe his (Sunak's) behaviour towards Boris Johnson, his disloyalty means that I could not possibly support him," minister for Brexit opportunities Jacob Rees-Mogg told Sky News on Wednesday.
Johnson, who won a large majority in December 2019, announced last week that he would step down after a torrent of resignations from ministers and with many Conservative Party lawmakers in open revolt.
His would-be successors have stressed how they would offer integrity and trustworthiness in contrast, but face questions of their own, such as why they backed Johnson for so long. Sunak was - like Johnson - fined for breaking lockdown rules.
Political opponents say the candidates have been focused only on winning the support of the right-wing of the ruling party by talking about tax cuts and extra defence spending while not addressing the cost of living crisis facing the public.
Polls also suggest that while the Conservatives argue among themselves, they are falling significantly behind the main opposition Labour Party, although no election is scheduled for a couple of years.
"I am your best shot of winning that election. I am the candidate that Labour fear," said Mordaunt as she officially launched her campaign on Wednesday.
People really wanted "the good old stuff" of low tax, small state and personal responsibility, she added.
Amid the promises, Andy King, board member of the Office for Budget Responsibility which independently monitors public finances, told a parliamentary committee taxes would have to rise or spending be cut to maintain fiscal sustainability.
The government has also said it will hold a confidence vote in itself on Monday - which it is almost certain to win - after blocking an effort to bring one by the opposition Labour Party.
