"We are extremely saddened to hear that Dr. Martin Roth has died. Martin will be remembered as a man of prodigious energy; a director with a global reputation both within the museum world and beyond," said Nicholas Coleridge, chairman of the celebrated design museum.
Roth, 62, died in his native Germany. He had been diagnosed with cancer after announcing his departure from the museum last year, according to British media reports.
The German foreign ministry said it was "profoundly saddened" by Roth's death, tweeting: "His commitment to fostering cultural collaboration across borders will be missed."
Roth led the museum for five years and was credited with raising its international profile.
He was behind successful exhibitions charting the lives of creatives such as David Bowie and Alexander McQueen.
Coleridge remembered Roth as "a committed Europhile and cultural ambassador with a philosophical turn of mind".
When he announced in September that he was stepping down, Roth said he feared Britain was becoming more introspective.
"I really like this country - I like London, I like to live here," he told BBC radio at the time.
However, "the terms and conditions are changing," he said.
"It's worse. The UK just started it now but this new nationalism is everywhere - it's a right-wing movement in Germany, it's in France, in the Netherlands, it's everywhere, and I think one has to do something, and that's one of the reasons why I'm leaving."
Roth went on to become president of the Institute for International Relations, based in Stuttgart in Germany, a post he formally started on July 1st.
"Our common path was too short," the Institute wrote in a Twitter post mourning Roth's death.
Before joining the V&A, Roth worked in Germany as director general of the State Art Collections in Dresden.