Mr Biden made the moving comments during a private memorial service at North Phoenix Baptist Church in Arizona.
The veteran centre-left politician began: “I’m Joe Biden. I’m a Democrat. And I love John McCain.
“I always thought of John as a brother”.
He explained how the two men met in the late 1970s, when Mr McCain was serving as a US Navy liaison officer to the Senate, whilst Mr Biden had already been elected.
According to Mr Biden the two men immediately “hit it off” and developed a “great friendship”.
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He added: “As a young man, he’d come up to my house, he’d come to Wilmington, and out of this grew a great friendship that transcended whatever political differences we had or later developed.
“Because above all, we understood the same thing. That all politics is personal. It’s about trust. And I trusted John with my life.
“We’d talk about politics. We’d talk about international relations. We’d talk about promise.”
Wiping away tears Mr Biden noted Mr McCain had been killed by the same type of brain cancer which took his son, Beau Biden, in 2015.
Mr Biden served as Vice President of the US between 2009 and 2017 under Barack Obama.
The Arizona memorial was also addressed by American football star Larry Fitzgerald, a friend of Mr McCain who plays for the Arizona Cardinals.
Mr Fitzgerald claimed: “What made Senator McCain so special was that he cared about the substance of my heart rather than where I came from.
“Whilst some might find our friendship out of the ordinary, it was a perfect example of what made him an iconic figure of American politics and service to fellow man.”
Mr McCain’s body will spend Friday lying in Washington D.C. in the US Capital building.
Members of the public will be allowed to pay their respects for a period of seven hours.
A second memorial service will take place on Saturday at the Washington National Cathedral.
Former Presidents Obama and George W. Bush will both make speeches at this event.
However President Trump, with whom Mr McCain was involved in a long-running spat, has not been invited.