In a first call lasting 7-8 minutes the two leaders discussed the fight against terrorism, the battle against the Islamic State group in Iraq and Syria, the conflict in eastern Ukraine and the Paris climate accord, a French presidential source told AFP.
The two men expressed a "desire to work together," the source added.
Climate change denier Trump has caused alarm in France by pledging to withdraw from the landmark deal to tackle global warming struck in Paris in December 2015.
But in their talks Hollande and Trump sought common ground, emphasising the friendship between their countries and the "history and values" they share, the source said.
Hollande, who is battling record low approval ratings ahead of presidential elections in France next year, had vowed a "frank" discussion with the Republican.
"Donald Trump has been elected. My duty is to ensure that we have the best relations but on the basis of frankness and clarity," Hollande told France 2 television earlier.
Hollande had made no secret of his desire to see Hillary Clinton win the White House, declaring a few months ago that Trump's excesses "make you want to retch".
His call with Trump came a day after talks between the forthcoming US president and German Chancellor Angela Merkel.
Merkel congratulated Trump and said she looked forward to meeting him, at the latest, when Germany hosts a G20 summit in July in the northern port city of Hamburg.
Merkel had offered Trump "close cooperation" and "stressed that Germany and the United States of America are closely tied through common values," her spokesman Georg Streiter said.
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