The company's Crew Dragon capsule, Endeavour, soared into the darkened pre-dawn sky atop a SpaceX Falcon 9 rocket shortly before 6 am Eastern time (1000 GMT) from NASA's Kennedy Space Center at Cape Canaveral, Florida, in a blastoff aired live on NASA TV.
One day's journey
The crew is due to arrive at the space station, which orbits nearly 250 miles (400 km) above Earth, early on Saturday following a flight of nearly 24 hours.
The mission marks the second "operational" space station team to be launched by NASA, aboard a Dragon Crew capsule since the United States resumed flying astronauts into space from US soil last year, following a nine-year hiatus at the end of the US space shuttle programme in 2011.
Mission number 3
It is also the third crewed flight launched into orbit under NASA's fledgling public-private partnership with SpaceX — the rocket company founded and owned by Musk, the billionaire entrepreneur who is also CEO of electric car maker Tesla Inc.
The first was an out-and-back test mission carrying just two astronauts into orbit last May, followed by SpaceX's first full-fledged four-member crew in November.
Friday's Crew-2 team consists of two NASA astronauts — mission commander Shane Kimbrough, 53, and pilot Megan McArthur, 49, — along with Japanese astronaut Akihiko Hoshide, 52, and fellow mission specialist Thomas Pesquet, 43, a French engineer of the European Space Agency.