Mr Collins was remembered as “a beloved father and grandfather” by his family, who said he died on Wednesday after “a valiant battle” with cancer.
They said he “spent his final days peacefully, with his family by his side”.
Mr Peake described him as an “inspiration and a legend”, adding: “Saddened to hear the passing of Michael Collins, who blazed a path for others to follow”.
Mr Peake launched into space in December 2015 as the first British ESA astronaut to visit the International Space Station.
He also retweeted a message from Mr Collins sent last week to mark Earth Day, which read: “I am certain, if everyone could see the Earth floating just outside their windows, every day would be #EarthDay.
“There are few things more fragile or more beautiful than Earth, let’s work together today and everyday to protect our home.”
Nasa said Mr Collins had “piloted humanity’s first voyage to the surface of another world” and that he “inspired generations”.
Mr Collins was critical to the mission 1969 mission as he piloted the ship from which Neil Armstrong and Buzz Aldrin left to make their historic first steps on the moon.
Mr Collins spent the eight-day mission piloting the command module.
He remained alone in the command module, Columbia, as his crew mates descended to the moon’s surface in the lunar lander.
None of the men flew in space after the Apollo 11 mission.
Mr Aldrin, 91, is now the only surviving member of the mission.
In tribute, he tweeted: “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.”
Mr Collins was alone for nearly 28 hours before his crew mates finished their tasks on the moon’s surface and lifted off in the lunar lander.
Had something gone wrong and Mr Aldrin and Mr Armstrong been stuck on the moon’s surface — a real fear — Mr Collins would have returned to Earth alone.