The decision, announced at a press conference in Berlin, covers priceless paintings and sketches by Picasso, Monet, Chagall and other masters that were discovered by chance in 2012 in the Munich flat of Cornelius Gurlitt.
Christoph Schäublin, president of the Board of Trustees at the the Museum of Fine Arts in Bern, pledged to work with German authorities to ensure that "all looted art in the collection is returned" to its rightful owners.
Gurlitt, who died last May aged 81, was the son of an art dealer tasked by Adolf Hitler to help plunder great works from museums and Jewish collectors, many of whom perished in the gas chambers.
In the course of a routine tax inquiry, 1,280 works were unearthed in Gurlitt's cluttered Munich flat.
More than 300 other works were discovered in a ramshackle home Gurlitt owned in Salzburg.
Although he was never charged with a crime, the German authorities confiscated all of the Munich pieces and stored them in a secret location.
Gurlitt struck an accord with the German government shortly before his death to help track down the paintings' rightful owners.
But his anger over his treatment reportedly led him to stipulate in his will that the collection should go not to a German museum but to the Swiss institution.
After six months of negotiations, German Culture Minister Monika Gruetters called the accord reached with the Bern museum "a milestone in coming to terms with our history" under the Third Reich.
She said the German government was committed to returning the looted works to Jewish descendants "as soon as possible, with no ifs, ands or buts".
But "we're at the beginning, not the end, of a long road," she admitted.
'Avalanche of lawsuits'
Had the Swiss museum unexpectedly turned down the offer, the pieces would have been divided up among relatives of Gurlitt, who never married and had no children.
Ronald Lauder, the head of the World Jewish Congress, declined to comment ahead of Monday's press conference.
But he told German news weekly Der Spiegel this month that the Swiss museum should not accept the inheritance, saying it "would open a Pandora's Box and cause an avalanche of lawsuits".
Underlining the point, one of Gurlitt's cousins, 86-year-old Uta Werner, said on Friday that she was contesting Gurlitt's fitness of mind when he wrote the will naming the Bern museum as his sole heir.
This could return the case to legal limbo, with ageing Jewish descendants left to fight for their claims in German courts for years to come.
After the discovery of the Gurlitt trove came to light in a magazine article last year, Jewish groups and the US and Israeli governments put pressure on Germany to establish a task force to investigate the works' provenance.
In the case of a Matisse painting found in the stash, called "Seated Woman" and believed to be worth around $20 million, the panel determined in June that the work was "Nazi loot" stolen from Paris art collector Paul Rosenberg.
His heirs include French journalist Anne Sinclair, former wife of ex-IMF chief Dominique Strauss-Kahn.
London lawyer Christopher Marinello, who represents the Rosenberg family, told AFP ahead of the decision that the museum — unlike individuals — would be bound by the Washington Principles, a 1998 international agreement on returning art stolen by the Nazis, as well as the 1986 International Council of Museums code of ethics.
"My clients have been extremely patient with German authorities throughout the process and enough is enough," he said.
Meanwhile the acquisition of the Gurlitt hoard would dramatically increase the prestige of the Bern institution, Switzerland's oldest art museum.
Stephan Klingen of Munich's Institute for Art History said the public interest in the collection was "enormous".
"I think this is a chance to show people right before their very eyes how problematic the handling of art and artworks after the war was," he told German news agency DPA.