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.
Stewart is the candidate with the lowest number of endorsements from fellow Conservative members of parliament but has waged a strong campaign on social media, reaching out to centre-ground voters from different parties.
Boris Johnson, the former foreign minister and leading Brexit campaigner, ran away with the first round of voting last week, winning the support of 114 MPs - more than twice the number of votes won by his nearest challengers.
Candidates to become Britain's next prime minister clashed over Brexit strategy at their first debate on Sunday but the front runner, Boris Johnson, dodged the confrontation.
The Cabinet Office Minister told a pro-Stewart gathering in London: "I think there is a yearning in this country for political leaders who tell it straight to people".
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"Thank you to those who supported me in the second ballot".
Five contenders in all were still in the running after a second-round ballot of Conservative lawmakers on Tuesday. "With his record as Mayor of London, Boris has shown the way to do this".
Simon Hoare, one of Mr Javid's backers, said he was "delighted" the home secretary had made it through to the next stage.
Raab's elimination from the contest after Andrea Leadsom and Esther McVey in previous rounds leaves Boris Johnson as the most prominent Brexiteer, a matter of concern for some.
Not all Mr Hancock's supporters will follow suit.
Mr Johnson is the frontrunner in the Tory leadership race which is in full swing.
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But the three other remaining candidates for the leadership said it would be economically damaging to leave the European Union abruptly if a deal was within Britain's grasp.
Additional votes will be held later this week among the legislators as necessary until the top two vote-getters remain.
"I may have been Culture Secretary but I don't have the oratory of Cicero, it's 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. "Very much look forward to taking part in tonight's BBC debate", he tweeted at 6.50pm.
His blunt rhetoric and buffoonish image have always been popular with party members but he is now ahead in the contest because MPs, previously much less enthusiastic about him, have come to see him as the candidate most likely to help them hang on to their seats.
It means Mr Johnson faces a tough examination in his first live televised debate this evening when he faces four MPs who have criticised his Brexit policy and his reluctance to take questions or campaign in public.
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