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."
It hasn't started well for Jeremy Corbyn, writes Carl Dinnen - but he has the advantage of expectations starting off extremely low.
Newly-elected Labour leader Jeremy Corbyn has finalised his shadow cabinet - so who has made the cut?
Shadow Business Secretary Chuka Umunna has left the Shadow Cabinet by mutual agreement, the MP has confirmed.