"My brother was a remarkable individual on every level," his sister Jody said in a statement.
"While most knew Paul Allen as a technologist and philanthropist, for us he was a much loved brother and uncle, and an exceptional friend."
Despite always staying in the shadow of Gates, his partner and friend since they were teenagers, Allen was a passionate lover of computers and a key figure in the creation of Microsoft in 1975.
"Paul Allen's contributions to our company, our industry and to our community are indispensable. As co-founder of Microsoft, in his own quiet and persistent way, he created magical products, experiences and institutions, and in doing so, he changed the world," CEO Satya Nadella said in a statement.
"I have learned so much from him - his inquisitiveness, curiosity and push for high standards are something that will continue to inspire me and all of us at Microsoft," he added.
According to Forbes magazine, Allen -- born in Seattle in 1953 and whose fortune amounted to some $21.7 billion -- was the world's 44th-richest person.
With the money he made from the software company, Allen -- together with his sister Jody -- founded the Seattle-based Vulcan Inc. conglomerate, which managed his private investments and his philanthropic activities.
"I have the honour to work at Microsoft because of the early work of Paul Allen. Paul has had immeasurable impact on the tech industry and the Seattle community. He will be deeply missed," tweeted company President Brad Smith.