Leyva, who won a bongo contest at the age of six and made his singing debut in 1932, had suffered a stroke on Sunday and died early on Thursday morning in hospital, his daughter Rosalia said.
The colourful improviser of traditional Cuban "son" music was the latest of the band's stars to die.
Compay Segundo, a guitarist, and Ruben Gonzalez, a pianist, died in 2003, aged 95 and 84. Ibrahim Ferrer, a singer, died last year at the age of 78.
The careers of the largely forgotten musicians were suddenly relaunched when they recorded a jam session with Ry Cooder in 1996 that became the award-winning Buena Vista Social Club album.
The recording rekindled interest in traditional Cuban music. Buena Vista was the name of a social club for older people in Havana.
The story of the band's late-life rise to international fame was told in the Oscar-nominated documentary of the same name by Wim Wenders, the German director.
Leyva, born in 1917 in Moron in central Cuba, had a deep, country voice and was well known by the 1950s for singing in the bands of Cuban greats Benny More and Bebo Valdez.
"Music was his life. He almost sang yesterday," Rosalia said at his wake on Thursday.