He told BBC News he had been a "bit surprised" by the scale of his victory but his campaign had showed "politics can change and we have changed it".
'Jez we did'
He will now select his shadow cabinet. Labour has confirmed Rosie Winterton will return as chief whip, but a string of existing cabinet members including Ms Cooper, Tristram Hunt and Rachel Reeves, have all ruled themselves out of serving on the front bench.
He has hinted that he wants to change the format of Prime Minister's Questions - he faces David Cameron across the dispatch box for the first time on Wednesday - suggesting other Labour MPs might get a turn.
And on Saturday night, he emailed party members asking them to submit questions the weekly exchange. "I want to be your voice," he wrote.
The Islington North MP won on the first round of voting in the leadership contest, taking 251,417 of the 422,664 votes cast - against 19% for Mr Burnham, 17% for Ms Cooper and 4.5% for Ms Kendall. Former minister and Gordon Brown ally Tom Watson was elected deputy leader.
Corbyn supporters chanted "Jez we did" as he took to the stage, putting on his glasses to deliver his acceptance speech.
The left-winger, who has spent his entire 32-year career in the Commons on the backbenches, promised to fight for a more tolerant and inclusive Britain - and to tackle "grotesque levels of inequality in our society".
He said the leadership campaign "showed our party and our movement, passionate, democratic, diverse, united and absolutely determined in our quest for a decent and better society that is possible for all".
"They are fed up with the inequality, the injustice, the unnecessary poverty. All those issues have brought people in, in a spirit of hope and optimism."
He said his campaign had given the lie to claims that young Britons were apathetic about politics, showing instead that they were "a very political generation that were turned off by the way in which politics was being conducted - we have to, and must, change that".
Mr Corbyn added: "The fightback now of our party gathers speed and gathers pace."
Addressing cheering crowds in Parliament Square, he delivered an impassioned plea to the government to recognise its legal obligations to refugees from Syria and elsewhere and to find "peaceful solutions to the world's problems".
"Open your your hearts. open your minds, open your attitude to suffering people, who are desperate and who are in need of somewhere safe to live," added the new Labour leader.
Singer Billy Bragg then led the crowd in a rendition of socialist anthem The Red Flag.
Mr Corbyn earlier told supporters his first day at the helm of his party in Parliament would be spent opposing government plans to "shackle" trade unions by imposing higher thresholds for strike ballots.
Analysis by BBC Political Editor Laura Kuenssberg
There are problems everywhere for Labour's new leader. He has always been an outsider, an insurgent in his own party.
How can he expect loyalty from his colleagues, unite the party, when he has rarely displayed it himself? MPs have been discussing ousting him for weeks. There is likely to be initial faint support from most. Don't expect a rapid coup.
But don't doubt most smiles behind him at the despatch box will be through gritted teeth. And shadow ministers' resignation letters have already been written.
An overwhelming 85% of people who signed up as affiliated supporters for £3 voted for Mr Corbyn - but he also topped the ballot among party members and trade unionists.
The BBC's assistant political editor Norman Smith said this broad support gave Mr Corbyn a strong mandate and would silence those on the right of the Labour Party who had been plotting to get rid of Mr Corbyn at the earliest opportunity, as he had "totally obliterated" his opponents.
Mr Corbyn's predecessor as Labour leader Ed Miliband gave his "full support" to Mr Corbyn and said he expected him to "reach out to all parts of our party" but ruled out a return to the front bench himself
Former Deputy Prime Minister Lord Prescott urged Labour frontbenchers thinking of resigning to think again, saying the party had "overwhelmingly endorsed" Mr Corbyn, who he said had got more votes than Tony Blair when he won the leadership in 1994.
"The party has spoken with a very strong voice. Get out and fight the Tories," Lord Prescott told BBC News.
Len McCluskey, general secretary of the UK's biggest union Unite, congratulated Mr Corbyn and Mr Watson, saying: "Voters can now look at Labour and see, unquestionably, that it stands for fairness, justice, peace and strong communities. It is the party of hope, ready to take on a Government hell-bent on making life worse for ordinary people."
SNP leader and Scotland's First Minister Nicola Sturgeon congratulated Mr Corbyn and offered to work with him to oppose the renewal of Trident nuclear weapons and against "Tory austerity".
But she added: "The reality today is that at a time when the country needs strong opposition to the Tories, Jeremy Corbyn leads a deeply, and very bitterly, divided party.
"Indeed, if Labour cannot quickly demonstrate that they have a credible chance of winning the next UK general election, many more people in Scotland are likely to conclude that independence is the only alternative to continued Tory government."
The prime minister spoke to Mr Corbyn on the phone to congratulate him on becoming the new leader of the opposition.
But Defence Secretary Michael Fallon, giving the Conservative Party's reaction, said: "Labour are now a serious risk to our nation's security, our economy's security and your family's security.
"Whether it's weakening our defences, raising taxes on jobs and earnings, racking up more debt and welfare or driving up the cost of living by printing money - Jeremy Corbyn's Labour Party will hurt working people."
Plaid Cymru leader Leanne Wood congratulated Mr Corbyn but said his election "cannot alter Labour's dismal record in government in Wales".
Green Party leader Natalie Bennett said: "The selection of Jeremy Corbyn, combined with the remarkable Green surge of the past year, and the SNP's success at the general election, shows how many people support an alternative to austerity economics."