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 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.
Paying tribute to Collins, Aldrin wrote in a tweet: "Dear Mike, Wherever you have been or will be, you will always have the Fire to Carry us deftly to new heights and to the future. We will miss you. May you Rest In Peace."