Anoushka Shankar, daughter of the late musician who died last week near San Diego, and her half-sister Grammy-winning singer Norah Jones, also paid their last respects at the service in a palm tree-lined meditation centre.
Tributes were read out from fellow musicians and artists who had been inspired by Shankar, labelled "The Godfather of World Music" by the Beatles and compared to Mozart by violin maestro Yehudi Menuhin.
Harrison, whose late husband learned sitar from Shankar and collaborated with him notably on the ground-breaking Concert for Bangladesh in 1971, said the former Beatle had learned so much from their friendship.
"They were like father and son as well as brothers... they made each other laugh as if they shared a secret. And I'm sure they did," said the 64-year-old, whose husband died of cancer in 2001.
Shankar "laid the stepping stones from West to East, that led George to new concepts, alternative philosophies and completely transformed his musical sensibilities," she said.
"They exchanged ideas and melodies until their minds and hearts, East and West, were entwined, like a double helix," she added in Encinitas, where Shankar had a home.
Shankar died last Tuesday at the age of 92, after failing to recover from surgery at a hospital in La Jolla, near San Diego.
Private memorial services were announced both in the United States and India, where Shankar also had a home.
Jazz-soul singer Jones, Shankar's daughter from an affair with a US concert producer, was dressed in black and kept a low profile at Thursday's event in Encinitas, up the coast from San Diego.
His widow Sukanya was also at the California memorial, which started with prayers chanted by M.N. Nandakumara of the Bharatiya Vidya Bhavan institute for Indian art and culture in London.
Nandakumara said that Shankar's music "brought people of various countries, communities together to his soul-stirring music, which was matchless.
"I do not know another musician who has understood the Eastern and Western music the way (Shankar) understood it, and interpreted it in such a way that people around the world were mesmerized by it," he said.
Grammys organizers the Recording Academy announced last week that Shankar, a three-time Grammy winner, is to receive a posthumous lifetime achievement award.
Shankar also received another Grammy nomination for the music industry's annual awards show, due on February 10, for "The Living Room Sessions Part 1," for Best World Music Album.