The Sheffield-born singer 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 rendition of With a Little Help From My Friends “sensational” and said: “He had a voice that was just unique.”
Wakeman told BBC Radio 2: “The great thing is with someone like Joe is what they leave behind, and that will be with us for years and years.”
Known for his gritty voice, Cocker – a former gas fitter – 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 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 visited 48 cities across the US.
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 2011.
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.
Cocker, who recorded 23 studio albums and 40 albums, lived in Colorado, in the US.
Mr Marshall said it was with “the heaviest hearts we heard that our beloved Joe Cocker passed away last night”.
He continued: “He was without the doubt the greatest rock/soul voice ever to come out of Britain and remained the same man throughout his life.”
Mr Marshall 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”.
The former Beatle said: “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.”
Other musicians to have paid tribute to Cocker include Beatles drummer Ringo Starr, who tweeted: “Goodbye and God bless to Joe Cocker.”
Aerosmith singer Steve Tyler tweeted: “We loved you forever, we will miss you always. RIP Joe Cocker.”
Another to pay tribute on Twitter was singer-songwriter Frank Turner, who wrote: “Wow. Sad to hear of Joe Cocker’s passing. Incredible singer.”
Bryan Adams tweeted: “Joe Cocker has died. RIP my good friend, you were one of the best rock singers ever.”
Meanwhile, 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”.