"I would respectfully say to my beloved European friends and colleagues that it's time that we snapped out of the general doom and gloom about the result of this election, and collective 'whinge-o-rama' that seems to be going on in some places," Johnson said at a press conference in Belgrade, Serbia, using British slang for complaining.
Johnson once said he was "genuinely worried that (Trump) could become president."
And after Trump claimed areas of London were dangerous due to radicalized Muslims, Johnson said: "The only reason I wouldn't visit some parts of New York is the real risk of meeting Donald Trump."
But he appears to have found a silver lining in Trump's win, saying the election was a "great opportunity for the UK" following Britain's seismic vote in June to leave the EU.
European leaders speak with Trump
Johnson's optimism contrasts with the lukewarm response from many EU politicians to Trump -- a candidate who lobbed insults at Europe and European leaders during his campaign, and was heavily criticized in turn.
Now they are having to put aside their differences with the President-elect and are pledging to work with him.
German Chancellor Angela Merkel, whom Trump slammed during the campaign for "ruining Germany"
by accepting large numbers of refugees, spoke on the phone Thursday with the President-elect, a German government representative told CNN.
Merkel congratulated Trump and pointed out the two countries have long-held common values, making them close.
Earlier, she had responded to Trump's win by saying the campaign had been notable for its confrontations, which had been hard to take.
On Friday, French President François Hollande -- who has said that Trump's comments on a dead soldier made him want to retch
-- had a "cordial" eight-minute phone conversation with the US President-elect during which they agreed to work to clarify positions on key issues, Elysée spokeswoman Elise Beretz told CNN.
Those issues included the fight against terrorism, Ukraine, Syria, Iraq and the Paris Agreement on climate change, she said.
They also "remembered the history and values that the two countries have in common, the amicable relations between France and the United States," Beretz said.
Hollande had earlier congratulated Trump on his win but said the result "leads to uncertainty," adding that he urged "vigilance because of statements made by Donald Trump."
EU chief sees 'new challenges'
British Prime Minister Theresa May also spoke to Trump on the phone Thursday in a call that ended with the President-elect inviting her to visit as soon as possible.
May had earlier been diplomatic in her congratulations, saying the two nations shared a "special relationship based on the values of freedom, democracy and enterprise."
"We are, and will remain, strong and close partners on trade, security and defense," she said.
European Council President Donald Tusk responded to Trump's win by saying he respected the democratic choice of the American people, but was "aware of the new challenges that these results bring."
One of those challenges, he said, was "this moment of uncertainty over the future of our trans-Atlantic relations."
"The events of the last months and days should be treated as a warning sign for all who believe in liberal democracy," Tusk said.
"This means that we should finally get our act together and bring back a sense of direction, bring back confidence, bring back a sense of order."
European right celebrates
But while the political establishment in Europe has reacted uneasily to Trump's win, parties on the right have rejoiced, likening it to the same mood of populist discontent they believe is redefining the political landscape on the continent.
Nigel Farage, the outgoing leader of the UK Independence Party and one of Brexit's top campaigners, told CNN that he and Trump had both triumphed in "the big battle of 2016." Farage, who spoke at a Trump rally in August, once called Trump "the new Ronald Reagan."
"I believe in nation-state democracy, I believe in controlling borders, I believe we've got to confront the threat of Islamic terrorism, and President-elect Trump believes in very much the same thing," he said from Miami.
But Farage said he considered it unlikely the British government would offer him a position as a mediator with Trump despite British press speculation about such a role.
Britain's ruling Conservative Party was "incredibly snobby about me, they find it very difficult to even have a conversation with me," he said.
Marine Le Pen, leader of France's far-right National Front,
which is riding high in polls ahead of next year's election, celebrated Trump's win with a flurry of tweets. "A new world is emerging," she wrote.
Florian Philippot, a National Front vice president, tweeted a photo of Le Pen with the caption: "Their world is collapsing. Ours is being built."