Mr Raab, who is currently housing minister, was a prominent Leave campaigner during the 2016 referendum.
Mr Davis quit late on Sunday night, saying Theresa May had "given away too much too easily".
The 44-year old Mr Raab, a lawyer before becoming an MP in 2010, will now take over day-to-day negotiations with the EU's Michel Barnier.
The UK is due to leave the European Union on 29 March 2019, but the two sides have yet to agree how trade will work between the UK and the EU afterwards.
There have been differences within the Conservative Party over how far the UK should prioritise the economy by compromising on issues such as leaving the remit of the European Court of Justice and ending free movement of people.
Mrs May's Conservative Party only has a majority in Parliament with the support in key votes of the 10 MPs from Northern Ireland's Democratic Unionist Party, so any split raises questions about whether her plan could survive a Commons vote - and has also led to renewed questions about whether she will face a challenge to her position.
Mr Raab has served in government since after the 2015 election, initially working in the Ministry of Justice before moving to the communities department in January.
The European Commission has declined to comment on the change of personnel, saying it would continue to negotiate with "good will" to try and secure an agreement on the terms of the UK's exit and future relations.
Asked how much of a problem Mr Davis's resignation was for the future of the negotiations, a spokesman replied: "It is not for us, we are here to work".
Mr Davis said he could not remain in his post because he no longer believed in the plan for the UK's future relations with the EU which was backed by the cabinet on Friday.
He said he hoped his resignation would make it easier for the UK to resist EU attempts to extract further concessions - but he insisted he was not seeking to undermine or challenge the prime minister.
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