Gillinson joined at the LSO as a cellist in 1970 and became managing director in 1984. He leaves this summer to become executive and artistic director of Carnegie Hall in New York.
John Tomlinson, a bass who sings regularly at the Royal Opera, the English National Opera, and the Bayreuth Festival, also received knighthoods.
Other musical figures on the Honours list included Joyce Foote, founder of the Chichester Music Festival, and soprano Lisa Milne, both of whom were named Members of the Order of the British Empire (MBE).
Financial Times dance critic Clement Crisp was named an Officer of the Order of the British Empire (OBE).