Khan, who is the first Muslim to occupy the top post, criticised British Prime Minister Cameron's government and his mayoral campaign rival Zac Goldsmith for their divisive election campaign.
The 45-year-old mayor, who is the son of a Pakistani bus driver and was elected by an overwhelming mandate earlier this week, said he wanted to focus on issues like transport and housing in London.
"They used fear and innuendo to try to 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," the Labour party MP said.
Throughout the mayoral campaign, the Tories were seen as trying to paint Khan as a radical, suggesting he had questions to answer because he had shared platforms with extremists in the past and defended them in his time as a human rights lawyer.
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"But there are two lessons in particular. First, Labour only wins when we face outwards and focus on the issues that the people actually care about. 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," Khan wrote.
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Khan, who is now being described as the most powerful Muslim politician in Europe, chose Southwark Cathedral for his swearing-in as mayor yesterday.
Khan, one of eight children born to Pakistani immigrants -- a bus driver father and a seamstress mother -- who grew up on a south London housing estate, has vowed to do all in his power to make London "better" in his new post and be a mayor for "all Londoners".
He would be stepping down as MP for Tooting to take charge of his new City Hall role, which will trigger a by-election in the east London constituency.