He died on Wednesday after "a valiant battle with cancer. He spent his final days peacefully, with his family by his side," they said.
Collins had stayed in lunar orbit as his colleagues Neil Armstrong and Buzz Aldrin walked on the Moon.
Aldrin, 91, is now the only surviving member of the mission.
In a statement, the Collins family said that "Mike always faced the challenges of life with grace and humility, and faced this, his final challenge, the same way".
"We will miss him terribly. Yet we also know how lucky Mike felt to have lived the life he did.
"We will honour his wish for us to celebrate, not mourn, that life."
The family also asked for privacy "during this difficult time".
On 16 July 1969, Neil Armstrong, Buzz Aldrin and Michael Collins were strapped into their Apollo spacecraft on top of the vast Saturn V rocket and were propelled into orbit in just over 11 minutes.
Four days later, Armstrong and Aldrin became the first humans to set foot on the lunar surface. Collins remained in the command module throughout the mission.
Armstrong's words, beamed to the world by TV, entered history: "That's one small step for man, one giant leap for mankind."