The 70-year-old is known for a number of hit songs including "America," "I'm a Believer" and "Sweet Caroline," which was written for presidential daughter Caroline Kennedy.
Diamond said he was "flying way above sea level" when heard about the honor.
"I've watched, and I've seen, and I've even dreamed that someday that would happen to me," he told The Associated Press. "But I never really believed that it would."
Streep, who has made more than 45 movies and has won two Oscars, said in a statement that she was deeply honored and wished her parents were alive to see it.
"All that education, allowance, tuition, voice lessons, summer jobs, scholarship application deadlines and loving care and discipline - all that they gave me, bore fruit in a way they never dreamed," she said.
Ma is one of the best-known classical musicians. He played for President John F. Kennedy and President Dwight D. Eisenhower at age 7 at a fundraiser for a national cultural center that would later become the Kennedy Center.
Rollins, who turned 81 on Wednesday, said the award was for the "great jazz artists that preceded me."
"I'm very happy that it's an honor for jazz because I think jazz is such an important spiritual force all over the world," he said.
Cook, who made her Broadway debut in 1951, said she cried when she received the letter informing her of the Kennedy Center Honors.
"It's a kind of validation for me, for my whole life," she said. "Because when I sing, I put my whole life, the good part, the bad part, I put that into my work."
Cook was cast in Voltaire's "Candide" and Meredith Wilson's 1957 hit musical "The Music Man." The Beatles later recorded the song she made popular, "Till There Was You."
President Barack Obama will salute the artists and others will perform in their honor at a ceremony at the Kennedy Center. CBS will broadcast the show Dec. 27.
The Associated Press contributed to this report.