The 46-year-old was taken to hospital earlier this month at the Walter Reed Army Medical Centre, near Washington.
Beau Biden was seen as a rising star of US politics but suffered health problems in recent years. He had intended to run for Delaware state governor in 2016.
“The entire Biden family is saddened beyond words”, the vice-president said.
In a statement, President Barack Obama said: “Michelle [Obama] and I are grieving tonight.”
“Beau was a good, big-hearted, devoutly Catholic and deeply faithful man, who made a difference in the lives of all he touched – and he lives on in their hearts.”
In a Twitter post, former President Bill Clinton wrote: “Hillary & I mourn the loss of Beau Biden – so full of life, love, honor, and service – and we pray for the strength of his wonderful family.”
“Beau embodied my father’s saying that a parent knows success when his child turns out better than he did,” he said, adding: “We know that Beau’s spirit will live on in all of us.”
“Beau Biden was, quite simply, the finest man any of us have ever known,” the statement ended.
In 2010, Beau Biden suffered a mild stroke. Three years later, he received surgery at a Texas cancer centre for what was said to be a small lesion.
The eldest of Mr Biden’s two sons, he served a year-long tour Iraq in 2008 as a captain in the Delaware National Guard.
As a child, Beau survived a car crash that killed his mother Neilia and his baby sister Naomi in 1972. He and his brother Hunter were injured.
Joe Biden had just been elected to the Senate, and took his oath of office by his son’s hospital bed.
Joe Biden married Jill Jacobs in 1977. They have one daughter – Ashley Blazer.
Beau Biden is survived by his wife Hallie and their two children, Natalie and Hunter.