Strummer, 50, collapsed and died after suffering a heart attack just days before Christmas. Family and friends bade their final farewell at a service at the crematorium as fans gathered outside in the driving rain.
The service was held in private at West London Crematorium, on Harrow Road.
Strummer left a widow, Lucy, two daughters, and a step-daughter.
They watched as the funeral cortege carrying Strummer's coffin passed what used to be The Elgin pub in Ladbroke Grove, west London, where The Clash played some of their earliest gigs 25 years ago.
Strummer, who was the band's guitarist, singer and, alongside Mick Jones, songwriter, died at his home in Somerset on December 22, a month after completing a tour with his latest project, The Mescaleros.
The family asked instead of floral tributes, money should be paid to the Mandela SOS fund-raising concert, which aims to raise awareness of the Aids epidemic in Africa.
The group's third album, London Calling, was named the greatest album of the 80s by music bible Rolling Stone despite being released in 1979.