In 1966, Surveyor 1 was the first US space probe to have a soft landing on the Moon.
The Soviet Union was the first when the Russian probe Luna 9 had a successful soft landing on the moon on February 3rd of that year. The Surveyor 1 mission was considered a complete success and demonstrated the technology necessary to achieve landing and operations on the lunar surface.
Learn more at http://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1966-045A