Hartwig Fischer, 53, who is currently director general of the Dresden State Art Collections in eastern Germany, will take up the post in the first half of 2016.
British Prime Minister David Cameron has confirmed the appointment ahead of incumbent Neil MacGregor's retirement in December, the museum said.
"I never dreamt that I would be invited to be responsible for this great British institution and I am conscious that nobody could fail to grasp what the British Museum represents not only for the UK but for the whole world," he said, adding "It's an honour".
The British Museum's chairman of trustees, Richard Lambert, hailed Fischer as "one of the outstanding museum directors in the world".
"He is not only a great scholar, but an experienced administrator and a gifted linguist with a global reputation for rethinking and representing great collections."
Linguist and scholar
Born in 1962 in the northern city of Hamburg, Fischer studied art history, history and archaeology in Paris, Rome, Berlin and Bonn, where he defended his thesis in 1994 on the German painter and sculptor Hermann Prell.
Fischer, who speaks four languages including English, French and Italian besides his native German, began his career in 2001 at Switzerland's Kunstmuseum in Basel where he was curator of 19th century and modern art.
In 2006, he was named head of Folkwang Museum in western Germany's Essen.
There he oversaw several major exhibitions, including one dedicated to art branded as "degenerate" by the Nazis, including works by Marc Chagall and Henri Matisse.
In May 2012, he took the reins of the State Art Collections in Dresden from Martin Roth, who also left for London, to head the Victoria and Albert Museum.
The 53-year-old Fischer's departure for London is "a real shock and a great loss" for the city and its cultural institutions, said the daily Die Welt, highlighting his academic and professional qualities.
He was a "control freak who wants a say on the tiniest detail," said the newspaper, which added that Fischer was "just as appreciated for his qualities as a serious researcher who treasures content, didacticism and new niche themes".
Even when he took the top job in Dresden, Fischer was a little known character beyond the arts world, with Die Welt then writing: "Hartwig who?"
But Wilhelm Krull, general secretary of the Volkswagen Foundation, which led the appointment committee, then said: "No other candidate knew as well as him how to link the 12 divisions of the museum together, to spark synergies for new exhibitions and cooperation."
The institution today receives about 2.5 million visitors annually.
That is dwarfed by the 6.7 million people who trooped to the British Museum last year.
But the former director of the London-based National Portrait Gallery, Sandy Nairne, told the Guardian newspaper that the quality of the Dresden collection is "refined, distinguished and vast".
And that is proof that Fischer is capable of leading an institution like the British Museum, added Nairne.
Fischer described himself as someone who "prioritizes action rather than words", while German news agency DPA called him "quiet, intellectual and a workaholic".
He had given a hint of his vision for a museum on International Museum Day in May, saying that the State Art Collections' "museums are social spaces".
"Spaces where artists, works of art, visitors and museum staff can encounter each other. In art, it is unthinkable to have boundaries or set limits."