The new mayor of London, who was signed in at a ceremony from which Mr Corbyn was notably absent, warned that appealing to "natural Labour voters" alone would not be enough to secure success, the Press Association reported.
Mr Khan secured an overwhelming victory over Conservative Zac Goldsmith following a bitter campaign which saw his rival attempt to link him to Islamic extremists.
The Labour mayor said the Tory campaign was "straight out of the Donald Trump playbook".
Mr Khan, who was congratulated by Mr Trump's probable opponent in the White House race, Hillary Clinton, said it was vital for Labour to be a "big tent" and avoid focusing on internal party issues.
Mr Corbyn's absence from Saturday's high-profile ceremony at Southwark Cathedral, in which Mr Khan was signed in as mayor, raised eyebrows and led to the party leader being forced to deny there was a rift.
The Labour leader's predecessor Ed Miliband was at the event, but Mr Corbyn's absence was unexplained.
Instead of appearing alongside Mr Khan, Mr Corbyn travelled to Bristol to congratulate that city's new mayor Marvin Rees. "I am meeting Sadiq over the weekend, I have been in touch with him. We are getting on fine. I have sent him a message of congratulations," he told reporters.
Mr Khan and Mr Corbyn are expected to have a meeting on Monday.
Writing in The Observer, the London mayor set out the lessons he had learned from his successful campaign.
"First, Labour only wins when we face outwards and focus on the issues that the people actually care about," he said.
"And secondly, we will never be trusted to govern unless we reach out and engage with all voters - regardless of their background, where they live or where they work.
"Squabbles over internal party structures might be important for some in the party, but it is clear they mean little or nothing to the huge majority of voters.
"As tempting as it might be, we must always resist focusing in on ourselves and ignoring what people really want."
He added: "Labour has to be a big tent that appeals to everyone - not just its own activists.
"Campaigns that deliberately turn their back on particular groups are doomed to fail.
"Just like in London, so-called natural Labour voters alone will never be enough to win a general election.
"We must be able to persuade people who previously voted Conservative that Labour can be trusted with the economy and security as well as improving public services and creating a fairer society."
Mr Khan said David Cameron and Mr Goldsmith had sought to "divide London's communities in an attempt to win votes".
"They used fear and innuendo to try and turn different ethnic and religious groups against each other - something straight out of the Donald Trump playbook. Londoners deserved better and I hope it's something the Conservative Party will never try to repeat."
Cabinet minister Michael Fallon said Mr Goldsmith's campaign in London, which has been criticised by senior Tories, was part of the "rough and tumble" of an election.
Mr Fallon described Mr Khan as a "Labour lackey who speaks alongside extremists" during the mayoral race, and was repeatedly challenged on the BBC Radio 4 Today programme to say whether he was worried about the capital's security with Mr Khan in City Hall.
"London is safe with a Conservative Government working with the new mayor of London," he said.
Tory former justice secretary Ken Clarke told the BBC the way the campaign had been run was a "mistake" and "probably had a counter-productive effect".
But Mr Fallon said: "Both candidates were asked questions about their backgrounds, their personalities, their judgment, the people they associate with. That's the nature of our democracy and the rough and tumble of politics."