British singers Lily Allen and Paloma Faith also paid tribute, with the latter describing his death as "a tragic loss".
"As if the week could get any worse," tweeted Allen, one of a number of celebrities to allude to this week's election results in America.
US comedian Sarah Silverman said his death would make people "remember songs like Come Healing which is a good one for these days".
Duran Duran singer Simon Le Bon wrote: "Spent a good deal of my early teens, just me & #LeonardCohen alone together in my suburban bedroom. It was a gentle & fulfilling love affair."
Alexandra Burke, whose cover version of Cohen's Hallelujah was the Christmas number one after she won The X Factor in 2008, said: "Leonard Cohen, a lyrical legend, a man who will continue to inspire. A voice that will always live on. Thank you for the memories."
Nick Cave, who covered Cohen's Avalanche on his 1984 album From Her to Eternity, said he was "the greatest songwriter of them all - utterly unique and impossible to imitate, no matter how hard we tried".
Neil Portnow, chief executive of the Recording Academy, which celebrates music through the Grammy Awards, described Cohen as "one of the most revered pop poets and a musical touchstone for many songwriters".
"His extraordinary talent had a profound impact on countless singers and songwriters, as well as the wider culture," he added.
"Unmatched in his creativity, insight and crippling candour, Leonard Cohen was a true visionary whose voice will be sorely missed," Cohen's manager Robert Kory told Rolling Stone.
"He leaves behind a legacy of work that will bring insight, inspiration and healing for generations to come."
BBC DJ Mark Radcliffe said Cohen had been "incredible to watch" when he performed at the Glastonbury Festival in 2008.
"There were 70,000 people in front of that stage but he made it feel like a tiny club," he recalled.
Like the late David Bowie, he went on, Cohen had "made great music right to the end with the enigma intact."
Producer John Lissauer, who worked with Cohen on such albums as New Skin for the Old Ceremony and Various Positions, said he had been "an iconic figure."