Johnson was appointed PM following May’s resignation earlier on Tuesday.
He formally took office after meeting Queen Elizabeth at the Buckingham Palace.
Following his meeting with the Queen, Johnson delivered a rousing speech outside No 10 Downing Street.
"The doubters, the doomsters, the gloomsters" who said it could not be done were "wrong", the new PM added.
He said he had "every confidence" the UK would leave the EU in 99 days time with a deal, but preparations for the "remote possibility" of a no-deal Brexit would be accelerated.
Johnson also pledged to improve infrastructure, recruit 20,000 new police officers and "level up" school spending.
Setting out his priorities for office, the former London mayor hit out at the "pessimists" who did not believe Brexit could be delivered and called for an end to three years of indecision.
"The people who bet against Britain are going to lose their shirts because we are going to restore trust in our democracy," he said.
"The time has come to act, to take decisions and change this country for the better."
Johnson also vowed to bring all four nations of the United Kingdom - or what he described as the "awesome foursome" - together in the task of strengthening a post-Brexit country.
"Though I am today building a great team of men and women, I will take personal responsibility for the change I want to see," he concluded.
"Never mind the backstop, the buck stops here."
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.
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 criticised.
Subscribe to Bishop Ikedi Blog
To receive alert from Bishop Ikedi Blog, Adverts Placement, News, Articles & more, Contact via Whatsapp: 07032803990, email firstname.lastname@example.org