Canadian-born Robertson toured with the veteran singer as the lead guitarist for his backing band in the mid-1960s, originally called the Hawks.
The group was later renamed The Band and Robertson helped write a number of their classic tracks such as The Weight, Up On Cripple Creek and The Night They Drove Old Dixie Down.
A statement posted on Dylan's Twitter, now rebranded as X, said: "This is shocking news. Robbie was a lifelong friend."
"His passing leaves a vacancy in the world," Dylan said.
The Band - Levon Helm, Rick Danko, Richard Manuel, Garth Hudson and Robertson - profoundly influenced popular music in the 1960s and 70s.
They received critical acclaim for their first two albums, 1968's Music From Big Pink and 1969's self-titled The Band.
In 1974, they appeared on Dylan's album Planet Waves and his 1975 record The Basement Tapes.
The group held a send-off concert in 1976 at San Francisco's Winterland Ballroom that included Dylan, Van Morrison, Neil Young and Muddy Waters among the line-up.
The concert was filmed by Martin Scorsese and was the basis for his celebrated documentary The Last Waltz, released in 1978.
Robertson also lent his musical talents to Scorsese's films including The Colour Of Money, The King Of Comedy, The Departed and The Irishman.
Rolling Stones rocker Ronnie Wood paid tribute to Robertson by sharing images from the rock documentary The Last Waltz, which he featured in.
"Such sad news about Robbie Robertson - he was a lovely man, a great friend and will be dearly missed," Wood said.
Neil Diamond, who produced Robertson's album Beautiful Noise, also paid tribute to the musician.
The American singer wrote: "The music world lost a great one with the passing of Robbie Robertson. Keep making that Beautiful Noise in the sky, Robbie. I'll miss you."
The Band was inducted into the Rock and Roll Hall of Fame in 1994.
Robertson died surrounded by family in Los Angeles "after a long illness", his publicist Ray Costa confirmed to the AP news agency.
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