The Sheffield born singer-songwriter had a career lasting more than 40 years with hits including You Are So Beautiful and Up Where We Belong.
His agent Barrie Marshall said Cocker, who died after battling lung cancer, was “simply unique”.
Sir Paul McCartney said he was a lovely guy who “brought so much to the world”.
Cocker’s friend Rick Wakeman, keyboard player for the rock band Yes, called his rendering of With a Little Help From My Friends “sensational” and said: “He had a voice that was just unique.”
“The great thing is with someone like Joe is what they leave behind, and that will be with us for years and years,” Wakeman told BBC Radio 2.
Known for his gritty voice, Cocker began his singing career in the pubs and clubs of Sheffield in the 1960s before hitting the big time.
He was propelled to pop stardom when his version of The Beatles’ With A Little Help from My Friends reached number one in 1968.
He performed the song at the famous Woodstock Festival in New York state a year later.
He was also well-known for his “Mad Dogs and Englishmen” Tour of 1970, which featured over 40 musicians touring 48 cities across the US, resulting in a third gold album of the same name and a concert film.
His duet with Jennifer Warnes, Up Where We Belong – from An Officer And A Gentleman – hit number one and went on to win both a Grammy and an Academy Award in 1983.
He was made an OBE in 2007.
Last year his arena tour across Europe saw him achieve a number one album in Germany and give what was to be his final concert in Hammersmith, London, in June.
Mr Marshall said it was with “the heaviest hearts we heard that our beloved Joe Cocker passed away last night”.
“He was without the doubt the greatest rock/soul voice ever to come out of Britain and remained the same man throughout his life.”
He described Cocker as a “true star” who was also “a kind and humble man who loved to perform”.
“Anyone who ever saw him live will never forget him,” he added.
‘Etched in our memories’
Sir Paul said he would be “forever grateful” to Cocker for turning With A Little Help From My Friends into a “soul anthem”.
“I knew him through the years as a good mate and I was so sad to hear that he had been ill and really sad to hear today that he had passed away.
“He was a great guy, a lovely guy who brought so much to the world and we’ll all miss him,” he added.
Among other musicians who have paid tribute to Cocker, was Beatles’ drummer Ringo Starr, who tweeted: “Goodbye and God bless to Joe Cocker.”
Singer song-writer Frank Turner tweeted: “Wow. Sad to hear of Joe Cocker’s passing. Incredible singer.”
BBC arts editor Will Gompertz said Cocker may have had “the finest blues soul voice ever to come out of Britain”.
Edgar Berger, chairman and chief executive of Sony Music Entertainment International, who signed Cocker, said he was “one of the most humble men I’ve ever met.
He said his voice “will forever be etched in our memories”.