The Stranglers' official website announced that Greenfield passed away on Sunday.
According to the statement, the musician tested positive for COVID-19 on April 26 after a prolonged hospital stay due to heart problems.
Greenfield, known for his immense contribution to Britain's punk scene in the late 1970s, joined The Stranglers in 1975.
His use of organ and synthesizers made the band different from the rest of the punk groups.
Greenfield also wrote the band's biggest hit, "Golden Brown", which released in 1982.
Other members of the band, guitarist and lead vocalist Baz Warne, bassist Jean-Jacques Burnel and drummer Jet Black, paid tributes to Greenfield on their website.
"We have just lost a dear friend and music genius, and so has the whole world. Dave was a complete natural in music. Together, we toured the globe endlessly and it was clear he was adored by millions. A huge talent, a great loss, he is dearly missed," Black wrote.
Hugh Cornwall, the band's former lead singer, said on Twitter, "I am very sorry to hear of the passing of Dave Greenfield. He was the difference between The Stranglers and every other punk band. His musical skill and gentle nature gave an interesting twist to the band."
Greenfield is survived by his wife, Pam.
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