The American businessman, who established the technology giant with Bill Gates in 1975, had been diagnosed with a recurrence of non-Hodgkin lymphoma earlier this month.
Mr Allen, whose shares in Microsoft made him one of the richest people in the world, was also known for his philanthropy – he gave away more than $1.5bn (£1.1bn) in his lifetime
An avid sports fan, the Seattle native owned the Seattle Seahawks NFL team and Portland Trail Blazers basketball team.
His family paid tribute to “a remarkable individual on every level”.
“While most knew Paul Allen as a technologist and philanthropist, for us he was a much-loved brother and uncle, and an exceptional friend,” they said in a statement issued through his company Vulcan Inc.
Mr Allen had announced on 1 October the cancer he battled in 2009 had returned. “I’ve begun treatment and my doctors are optimistic that I will see a good result,” he added.
Satya Nadella, Microsoft’s current chief executive, said: “Paul Allen’s contributions to our company, our industry and to our community are indispensable.”
He paid tribute to Mr Allen’s “inquisitiveness, curiosity and push for high standards” and said he would “continue to inspire me and all of us at Microsoft”.
Mr Allen persuaded Mr Gates to drop out of Harvard to start what would become the the world’s biggest software company. The two school friends’ big break came in 1980, when IBM moved into personal computers and asked Microsoft to provide the operating system.
To meet IBM’s needs, they paid $50,000 to buy a system known as as QDOS from another programmer, Tim Paterson. Eventually the refined product became the core of IBM computers and their clones, catapulting Microsoft into the domination position in the PC industry and thrusting Mr Allen and Mr Gates onto the throne of technology.
By 1991, Microsoft’s operating systems were used by 93 per cent of the world’s computers.
Mr Allen left Microsoft in 1983, following a dispute with Mr Gates, but his share of their original partnership made him a billionaire estimated earlier this year to be the 46th richest person in the world.
He spent his fortune on yachts, art, rock music, sports teams and real estate, as well as ocean conversation, homelessness and brain research.
In 1986, with his sister Jody, Mr Allen founded Vulcan, the investment firm that oversees his business and philanthropic efforts. He founded the Allen Institute for Brain Science and the aerospace firm Stratolaunch, which has built a colossal plane designed to launch satellites into orbit. He has also backed research into nuclear fusion power.
Mr Allen later joined the list of America’s wealthiest people who pledged to give away the bulk of their fortunes to charity. In 2010, he publicly pledged to give away the majority of his fortune, saying he believed “those fortunate to achieve great wealth should put it to work for the good of humanity”.
When he released his 2011 memoir, Idea Man, he allowed 60 Minutes inside his home on Lake Washington, across the water from Seattle, revealing collections that ranged from the guitar Jimi Hendrix played at Woodstock to vintage war planes and a 300ft yacht with its own submarine.
He was first diagnosed with cancer in 1983.
“To be 30 years old and have that kind of shock – to face your mortality – really makes you feel like you should do some of the things that you haven’t done yet,” he said in a 2000 book, Inside Out.
His influence is firmly imprinted on the cultural landscape of Seattle and the Pacific Northwest, from the bright metallic Museum of Pop Culture designed by architect Frank Gehry to the computer science centre at the University of Washington that bears his name.
Vulcan chief executive Bill Hilf said: “Paul’s life was diverse and lived with gusto. It reflected his myriad interests in technology, music and the arts, biosciences and artificial intelligence, conservation and in the power of shared experience – in a stadium or a neighbourhood – to transform individual lives and whole communities.
Additional reporting by agencies