Amber Rudd is Energy and Climate Change Secretary and Priti Patel will attend Cabinet as employment minister.
Boris Johnson will not be made a minister but will attend separate Tory "political cabinet" meetings.
Mr Cameron said Mr Johnson "would devote his attention, as promised" to his final year as Mayor of London.
The BBC's Norman Smith said Mr Johnson would be "integral" to Mr Cameron's team although it was an acknowledgement that he could not both "run London and a major government department".
Mr Cameron has been roundly applauded by backbench Conservative MPs as he spoke to the influential 1922 committee for the first time since winning a majority.
In other appointments announced on Monday:
- Sajid Javid moves to Business Secretary from Culture Secretary
- Veteran Tory MP John Whittingdale is to become Secretary of State for Culture, Media and Sport
- Baroness Stowell remains as leader of House of Lords and takes on the role of Lord Privy Seal
- Iain Duncan Smith remains as work and pension secretary
- Robert Halfon named as deputy chairman of the Conservative Party
A spokesman for Mr Johnson said he had accepted an invitation to attend weekly political cabinets - held after the full cabinet session - but would not take a ministerial role.
"The Mayor has always been clear - he has to fulfil his mandate running London first - and that's exactly what he will be doing until his term ends in May 2016," he said.
As he pieces together his new Cabinet, Mr Cameron is filling a number of top roles previously held by Lib Dems in the former coalition government.
Former energy minister Ms Rudd is promoted to take control of energy policy, a role held before the election by Ed Davey. Mr Javid, a rising star in the party, succeeds Vince Cable as business secretary.
Priti Patel takes the employment brief - previously held by Esther McVey
Sajid Javid moves from the culture department to business
As culture secretary John Whittingdale will lead negotiations on the renewal of the BBC's Royal Charter
Mr Johnson did not leave empty-handed - he will attend weekly "political cabinet" sessions
Mr Whittingdale, a former private secretary to Lady Thatcher, has joined the government for the first time. He was chair of the influential Commons culture committee over the past five years, overseeing its inquiry into phone hacking.
The Conservatives won a 12-seat majority in the House of Commons in Thursday's election, taking 331 of the 650 seats. You can see the full results here.
As MPs return to Westminster, Mr Cameron pledged to ensure the "economic recovery reaches all parts of our country".
Mr Cameron told the 1922 Committee his first term was "about repair and recovery", saying it fell to his party to put the economy back on track after "the great Labour recession", and that the next term is about renewal.
"It will be our task to renew a sense of fairness in our society - where those who work hard and do the right thing are able to get on."
On Europe, the prime minister said negotiations on reshaping the UK's membership "would be tough but we have a mandate".
The BBC's James Landale said Mr Cameron had also expressed frustration with pollsters for daily updates during the campaign suggesting the parties were level pegging, saying he was "going to sue them for my ulcers".
EU negotiations
Former EU Commission president Jose Manuel Barroso told Radio 4's Today programme he believed other EU leaders would be prepared to accommodate UK demands as long as they were "compatible with the overall project of European integration".
"The anti-European party, the Ukip party was reduced to a very small expression, almost irrelevant," he said.
"So today Prime Minister Cameron has internally much greater authority and from my experience - more than 10 years knowing him - I know he is someone determined and someone pragmatic."
However, he suggested some of the Conservatives' pre-election proposals - such as a four-year limit before EU migrants can claim certain benefits - could be "difficult" for other EU countries to accept.
Ministers from Poland, Hungary and Slovakia have told the Financial Timesthey are not prepared to give up on the principle of freedom of movement for their citizens.
In the first salvo over the EU referendum, which Mr Cameron has pledged to hold by the end of 2017, backbencher Graham Brady said not allowing ministers to campaign against continued membership could lead to tensions.
"Where you have people who have very, very strong beliefs about something, it is far better to recognise the strength of those views and to allow latitude and freedom of expression," he told BBC Radio 4's Westminster Hour.
Analysis by Europe editor Katya Adler
The first days in office are challenging for any new government and this will certainly be a busy week for David Cameron who, amongst other things, says he has already rolled up his sleeves to change the UK's relationship with Europe.
The first "fight" with Brussels already loomed large on Monday in the British papers.
The front-page lead story in The Times reported that "Brussels forces Britain to accept Med migrants". Migration is clearly a huge story, and a controversial one for the UK and Europe.
Ahead of Monday's reshuffle, it has already been announced Michael Gove will be the new justice secretary and Mark Harper the chief whip in Mr Cameron's cabinet while Nicky Morgan would remain as education secretary and minister for equalities.
David Cameron also confirmed on Friday that Chancellor George Osborne, Home Secretary Theresa May, Foreign Secretary Philip Hammond and Defence Secretary Michael Fallon would stay in their posts.
As work and pensions secretary, Mr Duncan Smith will now be able to continue his task of overhauling the welfare system and cutting its budget by £12bn.
He said: "I am delighted to have the opportunity to complete the vital welfare reforms that have already helped so many people back into work.
"The completion of Universal Credit will ensure work always pays, and will improve the incomes of those on low salaries."
In other news:
- The Parliamentary Labour Party will meet, as acting leader Harriet Harman warns against a "blame game" over its defeat
- UKIP's national executive committee will decide who will be interim leader of the party as Douglas Carswell rules himself out
- The BBC understands Angus Robertson will remain as SNP leader in the House of Commons as the party's new intake of 56 MPs are making their way to Westminster for the first time
- George Galloway says he has launched legal proceedings to challenge the result of the Bradford West election
- The British Chambers of Commerce has written to the PM urging him to take bold action to secure the UK's economic future
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