Armstrong, the first man to walk on the moon, died from complications following cardiovascular procedures, his family said in Ohio.
In a statement issued by the White House, the president said he and first lady Michelle Obama were saddened to hear about Armstrong's passing.
"Neil was among the greatest of American heroes -- not just of his time, but of all time," Obama said. "When he and his fellow crew members lifted off aboard Apollo 11 in 1969, they carried with them the aspirations of an entire nation. ... Today, Neil's spirit of discovery lives on in all the men and women who have devoted their lives to exploring the unknown -- including those who are ensuring that we reach higher and go further in space. That legacy will endure -- sparked by a man who taught us the enormous power of one small step."
Taking a break from the campaign trail in New Hampshire, Romney told reporters Armstrong's death was "very sad." In a subsequent statement, Romney said the first man to walk on the moon has a place in the "hall of heroes."
Romney said he met with Armstrong "a few weeks ago," and Armstrong's passion for space, science and discovery will inspire him for the rest of his life, and "the moon will miss its first son of Earth," The Huffington Post reported.