Armstrong, the first man to set foot on the moon on July 20, 1969, died on Saturday at the age of 82.
"Neil was among the greatest of American heroes, not just of his time, but of all time. When he and his fellow crew members lifted off aboard Apollo 11 in 1969, they carried with them the aspirations of an entire nation," Obama said, adding that he and the first lady are deeply saddened by his death.
"They set out to show the world that the American spirit can see beyond what seems unimaginable, that with enough drive and ingenuity, anything is possible," Obama said.
"And when Neil stepped foot on the surface of the moon for the first time, he delivered a moment of human achievement that will never be forgotten," Obama added.
"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," he said.
"That legacy will endure, sparked by a man who taught us the enormous power of one small step," he added.
The Obama statement came soon after the family of Armstrong announced his death.
"We are heartbroken to share the news that Neil Armstrong has passed away following complications resulting from cardiovascular procedures. Neil was our loving husband, father, grandfather, brother and friend," the family said in a statement.
"Neil Armstrong was also a reluctant American hero who always believed he was just doing his job. He served his Nation proudly, as a navy fighter pilot, test pilot, and astronaut," they said.
"He remained an advocate of aviation and exploration throughout his life and never lost his boyhood wonder of these pursuits," the family said.
"As much as Neil cherished his privacy, he always appreciated the expressions of good will from people around the world and from all walks of life," they added.
"While we mourn the loss of a very good man, we also celebrate his remarkable life and hope that it serves as an example to young people around the world to work hard to make their dreams come true, to be willing to explore and push the limits, and to selflessly serve a cause greater than themselves," the family said.
For those who may ask what they can do to honor Neil, the family said they have a simple request: "Honour his example of service, accomplishment and modesty, and the next time you walk outside on a clear night and see the moon smiling down at you, think of Neil Armstrong and give him a wink."
Armstrong inspired generations of Americans to believe that America is capable of achieving greatness that only comes with determination, perseverance, and hard work, US Defence Secretary Leon Panetta said.
"As a true pioneer, his one small step showed all mankind the great feats we can accomplish when we set ourselves to the task. While Neil is no longer with us, his spirit and his legacy of American achievement and national pride will live forever," Panetta said, paying tribute to the legendary US astronaut.
Mitt Romney, the Republican presidential candidate, said Neil Armstrong today takes his place in the hall of heroes.
"With courage unmeasured and unbounded love for his country, he walked where man had never walked before. The moon will miss its first son of earth," he said.
"I met and spoke with Neil Armstrong just a few weeks ago -- his passion for space, science and discovery, and his devotion to America will inspire me through my lifetime," Romney said.
NASA Administrator Charles Bolden said as long as there are history books, Armstrong will be included in them, remembered for taking "humankind's first small step on a world beyond our own".
Besides being one of America's greatest explorers, he carried himself with a grace and humility that was an example to all, Bolden said.
Democratic leader Nancy Pelosi said Armstrong was an American hero who never looked back, always moving the nation into new generations of exploration.
"When Neil Armstrong took his small step, Americans knew we could overcome any obstacle and prevail over any challenge. Throughout the rest of his life, he continued to work to inspire the next generation, to careers in science and innovation, to lives of service, to broad and bold perspectives," she said.
"Neil Armstrong was a true American hero, both because of his extraordinary service to his country and the honourable life he led. He was a groundbreaking naval aviator and the
world's most famous astronaut, but it was his humble and gracious response to the torrent of attention that followed his accomplishments that may have set him apart most," Ohio Senator Rob Portman said.