The Rolling Stones say they are devastated by the loss of their "very dear friend."
"Bobby made a unique musical contribution to the band since the 1960s. He will be greatly missed," The Stones said in a statement.
Bandmate Richards has also paid his own heartfelt tribute to Keys, with whom he was particularly close. In a handwritten letter posted on his Twitter page, Richards has spoken of his heartbreak.
"I have lost the largest pal in the world and I can't express the sense of sadness I feel, although Bobby would tell me to cheer up," wrote Richards, who was born on the exact same day as Keys - December 18, 1943.
"My condolences to all that knew him and his love of music."
Musician Leo Sayer has also hailed the Texas native, who played on the singer's hit song When I Need You.
"R.I.P. Bobby Keys," Sayer said in a statement.
"We toured together in the mid 70's, and Bobby was always fun to work with, a great musician everybody now knows as the sax man with the Rolling Stones. I'm so proud to have known you 'Texas'," Sayer said.
Keys began his music career as a teenager, touring with Buddy Holly and Bobby Vee when still in his teens.
• Read more: Rolling Stones saxophonist Bobby Keys dies
He befriended the Rolling Stones in 1964 and was later recruited to play on their 1969 album, Let It Bleed.
Keys was a mainstay on the British band's albums until 1974, then reunited with Mick Jagger, Richards and pals in 1980 for their Emotional Rescue opus.
He remained a key player on all subsequent albums, including 2005's A Bigger Bang.
Keys' biggest contributions were on hits such as Brown Sugar, in which he delivered a blasting solo, Can't You Hear Me Knocking and Happy.
He also played with Eric Clapton, Lynyrd Skynyrd and late Beatles stars John Lennon and George Harrison.
- AAP