He is being replaced at the international development department by Armed Forces minister Anne-Marie Trevelyan.
There is a return to government for former Defence Secretary Penny Mordaunt, who becomes paymaster general.
Meanwhile, Mr Cleverly is made a joint minister in the Foreign Office and Department for International Development.
Cabinet members remaining in place include Home Secretary Priti Patel; Foreign Secretary Dominic Raab; Cabinet Office minister Michael Gove; Health Secretary Matt Hancock, International Trade Secretary Liz Truss; Transport Secretary Grant Shapps; Defence Secretary Ben Wallace; Leader of the House Jacob Rees-Mogg; and Chief Whip Mark Spencer.
Mr Johnson is expected to appoint a new minister to oversee the building of the HS2 rail line, final approval for which was given this week.
Rishi Sunak, The New Chancellor
Arriving at the Treasury, Mr Sunak said he was "delighted to be appointed" chancellor and had "a lot to get on with".
The 39-year-old was educated at Winchester College and Oxford University, after which he went on to found an investment firm.
In 2015, he was elected MP for Richmond, North Yorkshire, replacing former Conservative leader William Hague.
Mr Sunak became a housing minister in 2018, before being promoted to chief secretary to the Treasury last July.
He stood in for Mr Johnson during the BBC's seven-way debate ahead of December's general election.
Commenting on Mr Javid's resignation, Labour's shadow chancellor John McDonnell said: "This must be a historical record with the government in crisis after just over two months in power.
"Dominic Cummings has clearly won the battle to take absolute control of the Treasury and install his stooge as chancellor."