The orchestra said Davis died Sunday after a short illness.
One of the best-known figures in British music, Davis worked with the London symphony for more than half a century.
He first conducted for the LSO in 1959 and took the principal conductor post in 1995, serving until 2006 before becoming president.
The orchestra said Davis had been "at the head of the LSO family for many years."
"His musicianship and his humanity have been cherished by musicians and audiences alike," it said in a statement, adding that "music lovers across the world have been inspired by his performances and recordings."
Associated in particular with the works of Wolfgang Amadeus Mozart, Jean Sibelius and Hector Berlioz, Davis won three Grammy awards — two in 2002 for the LSO's recording of "Les Troyens" by Berlioz, and one for Giuseppe Verdi's "Falstaff" four years later — and a host of other trophies.
Thanks to a generous relative, he studied at the private school Christ's Hospital, then at the Royal College of Music, before spending his compulsory military service as a clarinetist with the band of the Household Cavalry.
Because he did not play piano he was denied a place in the music college's conducting class, and initially he struggled to find conducting work. His entry in "Who's Who" listed the years 1949-57 as "freelance wilderness."
But after filling in to acclaim for Otto Klemperer at the Royal Festival Hall in 1959, his career took off.
Apart from his long association with the LSO, Davis spent periods as chief conductor of the BBC Symphony and music director of the Royal Opera House, and worked with ensembles around the world, including the New York Philharmonic, the Boston Symphony Orchestra and the Bavarian Radio Symphony Orchestra.
He was knighted by Queen Elizabeth II in 1980.
Antonio Pappano, music director of the Royal Opera House, said Davis' death was a blow to the company, which had planned to work with him again. Pappano said it "represents an end of an era, where grit, toil, vision and energy were the defining elements of a leading international opera house."
"The warmth and excitement of his music-making will be terribly missed. He was a giant," Pappano said.
In his early career Davis was, by his own admission, "a raw young man," known for abrasiveness and fits of temper. He mellowed with age, becoming an enthusiastic pipe-smoker and knitter as well as musical leader.
In 2007, he told the BBC that music helped stave off thoughts of death.
"Every time you give a concert, time is suspended: You're mastering it; time is not the enemy," he said. "It doesn't put off death, unfortunately, but it gives you a very good time while you're still alive."
Davis had two children with his first wife, April Cantelo, and five with his second, Ashraf Naini. His children survive him.
Copyright 2013 The Associated Press.