The veteran left winger got almost 60% of more than 400,000 votes cast, trouncing his rivals Andy Burnham, Yvette Cooper and Liz Kendall.
He immediately faced an exodus of shadow cabinet members – but senior figures including Ed Miliband urged the party’s MPs to get behind him.
Mr Corbyn was a 200-1 outsider when the three month contest began.
But he was swept to victory on a wave of enthusiasm for his anti-austerity message and promise to scrap Britain’s nuclear weapons and renationalise the railways and major utilities.
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”.
He will now select his shadow cabinet – but without a string of existing members including Ms Cooper, Tristram Hunt and Rachel Reeves – who have all ruled themselves out.
He has also hinted that he wants to change the format of Prime Minister’s Questions – he faces David Cameron across the despatch box for the first time on Wednesday – suggesting other Labour MPs might get a turn.
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 leftwinger, 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.”
His first act as leader was to attend a “Refugees Welcome Here” rally, joining tens of thousands of people marching through central London in support of the rights of refugees.
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.