The Archbishop of Canterbury called for "reconciliation" in the United States after the most bitter election campaign in modern history.
Justin Welby pledged to "pray for all the people of the United States" in the wake of Donald Trump's victory.
Following the result, Archbishop Welby said: "As President-elect Donald Trump prepares to take office, my continuing prayers are that the United States of America may find reconciliation after a bitter campaign, and that Mr Trump may be given wisdom, insight and grace as he faces the tasks before him."
His statement was muted and contained no message of congratulations.
Archbishop Welby will tomorrow host a live video discussion on Facebook on the subject of "Jesus calming the storm".
British Prime Minister Theresa May was among the first to congratulate the new president.
She said: "I would like to congratulate Donald Trump on being elected the next President of the United States, following a hard-fought campaign.
"Britain and the United States have an enduring and special relationship based on the values of freedom, democracy and enterprise.
"We are, and will remain, strong and close partners on trade, security and defence.
"I look forward to working with President-elect Donald Trump, building on these ties to ensure the security and prosperity of our nations in the years ahead."
Rock star Mick Jagger, who has objected to Donald Trump repeatedly playing his Rolling Stones hit You can't always get what you want at his rallies, tweeted:
Evangelical Christian blogger Ian Paul said: "The election of Trump is a stand against the relentless march of globalisation—just as was the vote for Brexit for many people—and the havoc it has wreaked in many parts of Western society. The impact of neo-liberalist economic policy is desperately divisive, creating a smaller and smaller elite acquiring greater and greater share of a nation's wealth, and creating a wider and deeper underclass of those who struggle and feel marginalised, and by and large destroying key structures within society, including traditional patterns of family relating."
Chine McDonald, director of communications at the Evangelical Alliance in the UK, tweeted:
In Israel, the right-wing minister Naftali Bennett tweeted that the era of the Palestinian state is now over:
Russia's President Vladimir Putin was also quick to welcome Trump.
He said in a congratulatory telegram to the President-elect that he looked forward to working with him on international affairs and to a "constructive dialogue" between the US and Russia in the interests of both nations.
In a statement, the Kremlin said: "Putin expressed hope for joint work to restore Russian-American relations from their state of crisis, and also to address pressing international issues and search for effective responses to challenges concerning global security."
Putin himself said on television later: "It won't be easy." But he wanted to restore "full relations" to the US and he wanted Russia to "do its part".
Other European leaders also welcomed the result.
Viktor Orban, the Hungarian prime minister who had backed Trump during the campaign, wrote on Facebook: "Congratulations! Very good news. We still live in a democracy."
Antoni Maczerewicz, the Polish defence minister and a fierce critic of Russia, said he expected an improvement in the already good Polish-US relationship. He added that Trump had stressed Poland was a close ally of the United States and "that the US never abandons its allies".
Jiri Ovcacek, spokesman for Milos Zeman, the Czech president, said "normal Americans have shown you can beat the lying media conglomerates".
But in France, Ségolène Royal, the environment minister, issued a warning to Trump that he cannot reverse the climate change deal reached in Paris last year to limit global warming. One of the promises Trump made during his campaign was to cancel it.
