Former Beatle Paul McCartney credited Cocker for giving "With a Little Help from My Friends," a mid-tempo melody, a whole new life.
"It was just mind blowing, totally turned the song into a soul anthem and I was forever grateful to him for doing that," McCartney said in a statement, calling the fellow Englishman a "lovely northern lad."
His explosive, at times unintelligible performance of that song at Woodstock helped him break through to larger audiences in what Rolling Stone magazine called "one of the most iconic sets from the legendary festival."
Cocker lived in Crawford, Colorado, and released nearly 40 albums as he toured the world during a career spanning five decades.
The Grammy winner was born into a working-class family and worked as a plumber while pursuing his singing career.
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He was named an Officer of the Order of the British Empire (OBE) at Buckingham Palace in 2011.
"Goodbye and God Bless to Joe Cocker from one of his friends peace and love. R.," tweeted Beatles drummer Ringo Starr.
In the early 1970s Cocker's "Mad Dogs and Englishmen" tour and live album cemented his success. In 1974, "You Are So Beautiful," co-written with Billy Preston, was a big hit. It was voted the fourth-most popular love song in a U.S. online poll in 2013.
In a memorable sketch in 1976, Cocker joined John Belushi as he parodied the singer's spasmodic style on the popular late-night comedy show "Saturday Night Live."
Cocker's career path was punctuated by struggles with alcohol and drugs.
In 1982, his career took off again with "Up Where We Belong," a duet with Jennifer Warnes for the film "An Officer and A Gentleman." The song earned Cocker his only Grammy and an Oscar for its writers.
His last studio album "Fire It Up" came out in 2012.
Cocker is survived by wife Pam, brother Victor, a stepdaughter and two grandchildren. He and his wife set up the Cocker Kids' Foundation to support local Colorado youth.