Mr Corbyn has won the leadership of the Labour Party by a landslide, taking almost 60 per cent of more than 400,000 votes cast.
But it comes amid internal party concerns that Mr Corbyn, a veteran left-winger with a history of defying the party whip, will be a divisive figure who will be considered unelectable by the British public.
Ms Sturgeon, Scotland’s First Minister, issued an immediate warning that Scotland’s appetite for independence may grow if Labour remains divided.
A YouGov poll for The Times today found Scotland remains sceptical about independence, but support has risen from 45 per cent at the referendum to 48 per cent and it comes on the back of other polls showing support for independence as high as 53 per cent.
Ms Sturgeon said: “We congratulate Jeremy Corbyn on his victory and hope that we can work constructively with him in a progressive alliance against Tory austerity.
“We also call on him to give an early commitment that Labour MPs will join the SNP in voting against the £100 billion renewal of Trident.
“However, 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.
“In the meantime, it is clearer than ever that the only credible and united opposition to the Tories, north and south of the border, is the SNP.”
In his victory speech, Mr Corbyn said “the fightback now of our party gathers speed and gathers pace” throughout the UK.
“I’m delighted that Kezia Dugdale is here today, our leader in Scotland,” he said.
“We’re all going to be campaigning in Scotland for those great Labour traditions in Scotland.”
In a statement, Ms Dugdale said: “Today shows politics has changed. People are calling for radical change and straight talk.
“Jeremy’s election shows that the party has listened to that call, and I look forward to working with him and meeting him to discuss his priorities later today.
“I have already said that I want people to take another look at the Labour Party. I want to say that again today. I hope that those who were lost to us in the past will start to listen again as both Jeremy and I put forward radical policies that we hope will win back support for Labour.
“We’ve now got the chance for a fresh start and new leadership in both Scotland and across the UK.
“I’ve said I want my leadership to be about shaking up the establishment in Scotland, and Jeremy wants to do the same across the UK. What people want is real change - not just in their politics, but in their lives. Today offers the chance for that change.”
Scottish Conservative leader Ruth Davidson said “many Labour colleagues are in despair” at Mr Corbyn’s appointment.
“Mr Corbyn has made clear that, under his leadership, Labour will flee the centre ground for a hard-left comfort zone which promotes policies of the 1970s.
“Even Andy Burnham has warned that by electing Jeremy Corbyn, the voting public will conclude that Labour has given up on being in government - and he’s right.
“This is especially true in Scotland where Jeremy Corbyn’s flirtation with the SNP will worry all two million of our country’s No voters.
“He refused to fight for our United Kingdom during the referendum because, in his own words, he was ‘doing stuff’ in London instead.
“I promise to always speak up for the millions of Scots - including hundreds of thousands of Labour supporters - who have always backed the United Kingdom as a source of solidarity.
“And it is the Scottish Conservatives who stand firmly in the modern centre ground of Scottish politics, fighting for improvements in our schools, our hospitals and our police force.
“The SNP seems unable to respect the result of last year’s referendum and is constantly agitating for a re-run.
“We need a strong Labour Party to stand behind the UK. So I am urging Mr Corbyn to come to Scotland this week and state unequivocally that he’ll do no deals with the SNP and will always fight to keep our United Kingdom together.”