77-year-old Starr, whose real name is Richard Starkey, is the second Beatle to be knighted, 21 years after Paul McCartney.
Gibb, the 70s disco pioneer who co-founded the BeeGees with his late brothers Robin and Maurice, said he was "deeply honored, humbled, and very proud" to be recognized.
"This is a moment in life to be treasured and never forgotten. I want to acknowledge how responsible my brothers are for this honor. It is as much theirs as it is mine," Gibb, 71, told the UK's Press Association.
Darcey Bussell, the former principal dancer with the Royal Ballet and current judge on hit British television show "Strictly Come Dancing," was awarded a damehood for services to dance.
More than 1,100 people were honored this year, including 551 women. The list, which is drawn up by the government and approved by the Queen, recognizes the achievements and service of "extraordinary" people and includes politicians, academics, journalists, authors and athletes.
The list includes five categories and is released twice a year -- once at New Year's and the other during the Queen's publicly celebrated birthday in June.
A knighthood or damehood is the highest honor. Men recognized as such are given the title "Sir," while women are given the title "Dame." Below that is a CBE, which stands for Commander of the Order of the British Empire; an OBE, or Officer of the Most Excellent Order of the British Empire; followed by an MBE, or Member of the Order of the British Empire.
Starr, McCartney, John Lennon and George Harrison were recognized with MBEs in 1965.
Actor Hugh Laurie, known for his roles in American shows such as the medical drama "House" and political satire "Veep," was recognized for services to drama, while former British Vogue editor Alexandra Shulman was recognized for her contribution to fashion journalism.
British author and journalist Jilly Cooper, who penned the racy romantic series "The Ruthshire Chronical," was honored for services to literature and charity.
Laurie, Shulman and Cooper were all awarded OBEs. They were previously honored with CBEs.