Boris Johnson has thanked voters in the north of England for "breaking the voting habits of generations" to back the Conservatives.
Speaking in Tony Blair's old seat of Sedgefield, the PM said he knew "how difficult" that decision can be.
Mr Johnson won a Commons majority of 80, his party's biggest election win for 30 years, by sweeping aside Labour in its traditional heartlands.
In contrast, Labour suffered its worst election result since the 1930s.
Activists chanted "Boris" as Mr Johnson arrived in the County Durham constituency, which returned a Conservative MP on Thursday for the first time in 84 years.
The prime minister said he wanted to thank voters in the "incredible" constituencies in north-east England for placing their trust in the Conservatives.
They had "changed the political landscape" and "changed the Conservative Party for the better", he said.
"Everything that we do, everything that I do as your prime minister, will be devoted to repaying that trust," Mr Johnson added.
"We are the servants now and our job is to serve the people of this country and deliver on our priorities. And our priorities and their priorities are the same."
Leader Jeremy Corbyn said he had done "everything I could" to get Labour into power but expected to stand down "early next year", after a successor has been chosen by the party.
He said the general election had been "taken over by Brexit", the issue on which Mr Johnson campaigned most vociferously - but other figures in the party have disagreed over the reason.
Shadow chancellor John McDonnell promised to "learn lessons and we'll listen to people" during the debate over the future of the party and its next leader.
"My fear is that we're in for the long haul now, possibly five years," he added.
Labour's Helen Goodman, who lost the seat of Bishop Auckland to the Conservatives, told BBC Radio 4's Today programme that "the biggest factor" in Labour's defeat "was obviously the unpopularity of Jeremy Corbyn as the leader".
However, the Labour MP for York Central, Rachel Maskell, said: "We've all got to take responsibility... I don't think apportioning blame to a complex situation in a simplistic way is really the way to approach this."
Mr Johnson is expected to announce a minor government re-shuffle as early as Monday.
Asked whether his promise to be a one nation government meant bringing back Tory politicians like Penny Mordaunt and Jeremy Hunt - who left cabinet in July after Mr Johnson took over - the PM said he was "not going to speculate about personalities".
MPs will then return to Westminster on Tuesday and begin the process of swearing in, before the Queen formally opens Parliament on Thursday with "reduced ceremonial elements".
The prime minister has also vowed to reintroduce his Withdrawal Agreement Bill to Parliament before Christmas, which could happen by the end of next week.
It would see MPs begin the process of considering legislation that would pave the way for the UK to leave the EU on 31 January. Talks about a future trade and security relationship will begin almost immediately.
Former Conservative Deputy Prime Minister Lord Heseltine, who opposes Brexit and backed the Liberal Democrats in the election, told Today: "We've lost. Brexit is going to happen and we have to live with it."
Asked whether he would support any future campaign to rejoin the EU, he said it would be "20 years or something before the issue is once again raised".
Protests took place at Westminster on Friday following Mr Johnson's election victory.
Demonstrators in Westminster carried signs that read "Defy Tory Rule" and "No to Boris Johnson".
The Metropolitan Police said two people had been arrested in relation to the protests - one person on suspicion of assaulting a police officer and another for suspected affray.
Following the Conservatives' election win, Mr Johnson spoke to SNP leader Nicola Sturgeon on Friday evening and reiterated his opposition to a second independence referendum in Scotland.
The conversation came after the first minister said the PM had "no right" to stand in the way of a second vote following her party's "overwhelming" election performance. The SNP won 48 of Scotland's 59 seats.
Speaking on Radio 4's Any Questions on Friday, cabinet minister Thérèse Coffey insisted there would be no referendum on Scottish independence during the Conservatives' five-year term.
After speaking to Ms Sturgeon, the PM also took phone calls from German Chancellor Angela Merkel and Irish Prime Minister Leo Varadkar to discuss the next steps on Brexit.
The Conservatives won a total of 365 seats in the election, while Labour finished on 203, the SNP on 48, Liberal Democrats on 11 and the DUP on eight.
Sinn Fein has seven MPs, Plaid Cymru four and Northern Ireland's SDLP two. The Green Party and NI's Alliance Party have one each.
The Brexit Party - which triumphed in the summer's European Parliament elections - failed to win any Westminster seats.
The Conservatives swept aside Labour in its traditional heartlands in the Midlands and the north of England and picked up seats across Wales, while holding off the Lib Dem challenge in many seats in the south of England.
Voter turnout overall, on a cold and damp polling day, was 67.3%, which is down by 1.5% on the 2017 total.
Meanwhile, the Liberal Democrats are looking for a new leader after Jo Swinson lost her Dunbartonshire East seat to the SNP by 149 votes.
While she admitted her "unapologetic" pro-Remain strategy had not worked, she said she did not regret standing up for her "liberal values" and urged the party to "regroup and refresh" itself in the face of a "nationalist surge" in British politics.
Sir Ed Davey and Baroness Sal Brinton will be acting co-leaders for the party now that Ms Swinson is no longer an MP.