Clemons died on Saturday night in hospital where he had been following a stroke last week at his home in Singer Island, Florida.
Springsteen called the loss "immeasurable".
"We are honoured and thankful to have known him and had the opportunity to stand beside him for nearly 40 years," Springsteen said on his website.
"He was my great friend, my partner, and with Clarence at my side, my band and I were able to tell a story far deeper than those simply contained in our music.
"His life, his memory and his love will live on in that story and in our band."
Known as the Big Man for his imposing 6ft 5in, 19 stones frame, Clemons and his ever-present saxophone spent much of his life with The Boss, and his booming saxophone solos became a signature sound for the E Street Band on many key songs, including Jungleland, a triumphant solo he spent 16 hours perfecting, and Born To Run.
Outside The Stone Pony, the legendary rock club in Asbury Park, New Jersey, where Springsteen, Clemons and other E Street Band members developed their craft in the 1970s, Phil Kuntz stopped to place a small yellow flower on a decorative white fence. Nearby, someone taped a handwritten sign that read simply "RIP Big Man".
"I'll never hear Jungleland played live again, and that's a bummer," said Mr Kuntz, 51, who had seen Clemons perform with Springsteen more than 200 times.
In recent years, Clemons had been slowed by health woes. He endured major spinal surgery in January 2010, and, at the 2009 Super Bowl, Clemons rose from a wheelchair to perform with Springsteen after double knee replacement surgery.
But his health seemed to be improving. In May, he performed with Lady Gaga on the final show of American Idol, and performed on two songs on her Born This Way album.
Clemons said in a 2010 interview that he was winning his battles - including severe chronic pain and post-surgical depression.
"Of all the surgeries I've had, there's not much left to operate on. I am totally bionic," he said.
"God will give you no more than you can handle," he said in the interview. "This is all a test to see if you are really ready for the good things that are going to come in your life. All this pain is going to come back and make me stronger."
An original member - and the oldest member - of the E Street Band, Clemons also performed with The Grateful Dead, The Jerry Garcia Band, and Ringo Starr's All Star Band.He recorded with a wide range of artists including Aretha Franklin, Roy Orbison and Jackson Browne. He also had his own band called The Temple of Soul.
The stage "always feels like home. It's where I belong," Clemons, a former youth counsellor, said after performing at a Hard Rock Cafe benefit for Home Safe, a children's charity, in 2010.
Born in Norfolk, Virginia, Clemons was the grandson of a Baptist minister and began playing the saxophone when he was nine.
"Nobody played instruments in my family. My father got that bug and said he wants his son to play saxophone. I wanted an electric train for Christmas, but he got me a saxophone. I flipped out," he said in a 1989 interview.