May won 199 votes and Leadsom 84 in a second ballot of MPs of the governing Conservative party on Thursday.
Justice Secretary Michael Gove took just 46 votes and was eliminated from the race.
"This vote shows that the Conservative Party can come together, and under my leadership it will," May told supporters after the results were announced.
Around 150,000 grassroots Conservatives across the country will now vote to decide whether May or Leadsom becomes Britain's first woman prime minister since Margaret Thatcher was forced from office in 1990.
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The result of the contest is expected by September 9, meaning businesses and investors must endure two more months of uncertainty over who will lead the huge task of disentangling Britain's economy from the EU while trying to safeguard trade and investment.
Prime Minister David Cameron said last month he was stepping down after voters, many of them swayed by concerns over high immigration and a desire to reclaim "independence" from Brussels, rejected his entreaties to keep Britain in the EU and his warnings that leaving would spell economic disaster.
"This vote shows that the Conservative Party can come together, and under my leadership it will," May told supporters after the results were announced.
Until a couple of weeks ago Leadsom, a junior energy minister, was barely known to most Britons, but as one of the leading voices in the successful Leave campaign she has dramatically emerged as a serious challenger as better-known rivals have been felled by political intrigue.
"She's the future. What you want is someone who's not been in the Westminster bubble, she's had a real job in the real world and that's what people are looking for. I think they're fed up with the political elite," said Leadsom backer Peter Bone.
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Interior minister May, 59, has served for the past six years in one of the toughest portfolios in government. Leadsom has never been a member of the cabinet.
But despite her strong lead in the vote of MPs, May is far from assured of winning the race for Downing Street. During the referendum campaign she took a low-key stance in favour of Britain remaining in the EU - unlike Leadsom, who fought on the winning side.
Grassroots party members have strong eurosceptic leanings, a factor that could favour Leadsom.
The 53-year-old, who entered parliament only six years ago, said on Thursday her top priority would be to guarantee tariff-free trade with the EU after leaving.
But the EU is likely to insist that this would only be feasible if Britain continued to allow other EU nationals to live and work freely in Britain, an arrangement that has pushed immigration to record levels and was a powerful factor behind the success of the Leave side in the referendum.
