Mr Obama said on his Twitter feed: "Neil Armstrong was a hero not just of his time, but of all time."
Hundreds of millions watched Armstrong land on the Moon on 20 July 1969 and describe it as: "One small step for a man, one giant leap for mankind."
The line became one of the most famous quotes of the 20th Century.
Armstrong's family confirmed his death in a statement on Saturday, saying he had died from complications after surgery to relieve four blocked coronary arteries.
The family statement praised him as a "reluctant American hero" and urged his fans to honour his example of "service, accomplishment and modesty".

Mr Obama thanked Armstrong for showing the world "the power of one small step".
'Nerdy engineer' Last November he received the Congressional Gold Medal, the highest US civilian award.
Many of Armstrong's colleagues and friends paid tribute to him as a modest, private man who never sought the limelight.
Michael Collins, a pilot on the Apollo 11 Moon mission, said: "He was the best, and I will miss him terribly."

Armstrong famously refused most public appearances and interviews. In a rare interview with Australian TV this year, he reflected on a moment during his three hours on the Moon when he stopped to commemorate US astronauts and Soviet cosmonauts who had died in action. "It was special and memorable, but it was only instantaneous because there was work to do," he said. More than 500 million TV viewers around the world watched its touchdown on the lunar surface. Armstrong and fellow astronaut Edwin "Buzz" Aldrin collected samples, conducted experiments and took photographs during their moonwalk. Mr Aldrin told the BBC he would remember his colleague as a "very capable commander and leader of a world achievement". "We're missing a great spokesman and leader in the space programme," he said. Apollo 11 was Armstrong's last space mission. In 1971, he left the US space agency Nasa to teach aerospace engineering.
Born in 1930 and raised in Ohio, Armstrong took his first flight aged six with his father and formed a lifelong passion for flying.
He flew Navy fighter jets during the Korean War in the 1950s, and joined the US space programme in 1962.
Correspondents say Armstrong remained modest and never allowed himself to be caught up in the glamour of space exploration.
"I am, and ever will be, a white-socks, pocket-protector, nerdy engineer," he said in February 2000 in a rare public appearance.
Nasa chief Charles Bolden paid tribute to him as "one of America's great explorers".
"As long as there are history books, Neil Armstrong will be included in them, remembered for taking humankind's first small step on a world beyond our own."