Former deputy prime minister John Prescott, who served under Tony Blair, warned Mr Corbyn leading the party "won't be easy".
Writing in the Mirror on Sunday, he said: "This contest has been a bit bitter at times and focused more on personalities than policies. Bridges have been burned and will need to be rebuilt. Those you opposed you'll have to embrace.
"And those who opposed you will need to swallow their pride and realise you are the most democratically elected leader we have ever had. No-one can take that mandate from you.
"You thoroughly deserve our support and the opportunity to lead Labour and her Majesty's Opposition. Now let's come together to hold the Tories to account and win in 2020."
But Labour backbencher Simon Danczuk said Corbyn would need to pass "a series of crucial tests to convince his MPs that our aim is to win the next general election- not indulge in some ideological experiment.
Writing in the Mail on Sunday, he added: "He must outperform Prime Minister David Cameron in the Commons. We cannot have a Labour leader introduced to gales of laughter every week."
Jeremy Corbyn has said that Rosie Winterton, will remain in her role as Labour's chief whip, as he draws up his shadow cabinet.
"Rosie has served our party under consecutive administrations and will be a valuable part of our new team," Mr Corbyn said, whose victory as Labour leader sparked a flurry of resignations within the party.
Ms Winterton, a former aide to John Prescott and a minister under Tony Blair and Gordon Brown, held the chief whip role throughout Ed Miliband's leadership of the party.
Yvette Cooper has missed out on being leader of the Labour party after Jeremy Corbyn stormed to victory. The Normanton, Pontefract and Castleford MP finished third in the end, behind Andy Burnham. Some of our MPs - including Yvette Cooper - have already refused to be part of Jeremy Corbyn's top team, saying he's too left-wing, There is some flash photography in our Political Correspondent Paul Brand's report.
MPs Andy Burnham and Yvette Cooper have tweeted their congratulations to new Leader of the Opposition Jeremy Corbyn.
Mr Corbyn won with nearly 60 per cent of the vote. Mr Burnham came second with 19 per cent, and Yvette Cooper third with 17 per cent.
Rachel Reeves MP, the shadow work and pensions secretary, has announced she will leave Labour's frontbench following Jeremy Corbyn's election as Labour leader.
The MP for Leeds West is currently on maternity leave, but confirmed over Twitter she would be stepping down when she returns to work. She also tweeted her congratulations to Mr Corbyn and Labour's new deputy leader, Tom Watson MP.
Former leader of the Labour party Ed Miliband has ruled himself out of serving in new leader Jeremy Corbyn's frontbench team.
Asked if he would serve in a shadow cabinet under Mr Corbyn, he said they had discussed it, but he would be staying on the backbench.
He said: "I believe the right thing for me to do is to be a backbencher, serving my constituency and serving the issues I care about, supporting him on issues I care about, equality and climate change, and that's what I intend to do."
On the election result, he said the party "has made its choice and I think that choice should be respected."
He said the surge in members and affiliated supporters was "a massive opportunity" for Labour and said the challenge was to turn the half a million supporters and members into an "active" part of the party.
Jeremy Corbyn has been elected as the new leader of the Labour party, with 59 per cent of the vote.
Tom Watson has been elected as the new deputy leader of the Labour party.
He won 50.7 per cent in the third round, with 198,962 votes.
In the first round, he won 160,852 votes, representing 39.4 per cent. Ben Bradshaw was eliminated after coming lowest.
In the second round, Angela Eagle was eliminated. Watson won 170,589 - 42.2 per cent - of the total 408,470 votes cast in the election.
Labour members are gathered at the Queen Elizabeth II conference centre in London, where the result of the leadership election is due to be declared shortly.
The results of the election to select the new Labour party leader and deputy leader are being announced today.
*This live broadcast has now finished\*
Turnout in the Labour leadership election was 76.3 per cent, according to party sources.
Around 550,000 people were registered to vote.