Speaking outside No 10, he said Brexit would happen on 31 October “no ifs, no buts”, adding “the buck stops here”.
He has named Sajid Javid as his new chancellor while Priti Patel returns to government as home secretary.
More than half the previous cabinet, including leadership rival Jeremy Hunt and many of his supporters, have quit or been sacked.
Announcing his departure, the foreign secretary said he had been offered an alternative role but had turned it down.
Defence Secretary Penny Mordaunt, a leading Brexiteer who is popular across the party, was one of the more surprising of the 16 departures so far.
Another prominent Brexiteer, International Trade Secretary Liam Fox, was also ousted, along with Business Secretary Greg Clark – a vocal opponent of a no-deal Brexit.
All three supported Mr Hunt in the Tory leadership contest.
Education Secretary Damian Hinds, Northern Ireland Secretary Karen Bradley, Immigration Minister Caroline Nokes, Culture Secretary Jeremy Wright and Communities Secretary James Brokenshire have also gone, along with Chris Grayling, whose record as Transport Secretary was much criticed
This comes on top of the earlier resignations of four leading ministers, including Chancellor Philip Hammond, Justice Secretary David Gauke and Cabinet Office minister David Lidington.
Conservative MP Nigel Evans described the changes as a “summer’s day massacre”.
The BBC’s chief political correspondent Vicki Young said the sackings suggested Mr Johnson wasn’t looking to build bridges across the party.
Instead, she said, he was focused above all else on assembling the team he thought would bring about the results he needed, even if that was controversial.
Former Home Secretary Sajid Javid – a banker before entering politics – has been given the key role of chancellor, having thrown his weight behind Boris Johnson after being eliminated from the leadership race himself.
Priti Patel – who was sacked as international development secretary in 2017 for holding unauthorised meetings with Israeli officials – succeeds Mr Javid at the Home Office.
Dominic Raab – a former Brexit secretary – is also tipped for a top job while Michael Gove, Liz Truss and Steve Barclay have also been seen going into Downing Street.