The global development minister Rory Stewart was fourth on 37 votes and the interior minister Sajid Javid was fifth on 33 votes, as hardline Brexiteer Raab was eliminated after he failed to receive the required minimum of 33 votes.
Rory Stewart, the outsider at the start of the process, gained 18 votes to go through on 37.
The final two names will then be put to a postal vote of the 160,000 Tory party members, beginning on 22 June, with the victor expected to be announced about four weeks later.
Additional votes will be held later this week among the legislators as necessary until the top two vote-getters remain.
Boris Johnson has increased his lead in the race to become Britain's next prime minister, as one of his rivals was eliminated from the contest.
But the speaker of the House of Commons, John Bercow, said it was fantasy to think that the lower house of parliament could be pushed aside.
"He is the best placed to get us out of the European Union at the end of October", Leadsom, a former leader of the House of Commons who was eliminated from the contest last week, told LBC radio.
"His view and mine is very closely aligned there".
Mr Johnson won the second round of the contest - announced at 6pm on Tuesday - with the backing of 126 out of 313 Tory MPs.
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Scottish Secretary David Mundell switched to the Gove camp instead.
Former Brexit Secretary Dominic Raab, who resigned from the government in protest at the government's compromises with Brussels, went further than the others in not ruling out the suspension of parliament if needed to stop MPs from blocking a no-deal Brexit.
But elsewhere across the Black Country and Staffordshire there was support for every candidate.
"I say as far as I can see because he doesn't talk to me, he doesn't talk to you, he doesn't talk to the public", said Stewart, who is now placed second favorite.
Eddie Hughes, the MP for Walsall North, had backed Mr Raab, describing him as "the only candidate with a credible plan to get us out of the European Union by October 31".
Tonight's result meant Kent MPs Rehman Chishti, Helen Grant and Gareth Johnson would have to find another candidate to support.
Ahead of the second round of voting he described himself as "less Homer's Iliad and more Homer Simpson".
Mr Javid, the state-school educated son of a Pakistani bus driver, has a background which is in stark contrast to rivals including Old Etonians Boris Johnson and Rory Stewart.
Home Secretary Sajid Javid just made the minimum vote cut-off of 33, while surprise packet Rory Stewart, the International Development Secretary, almost doubling on his 17 votes from the last ballot to 37 on Tuesday's.