An earlier launch of the Falcon 9 rocket was aborted early Saturday in the final seconds before liftoff from Cape Canaveral.
The real test comes Thursday when the Dragon gets close to the space station. It will undergo practice maneuvers from more than a mile out. If all goes well, the docking will occur Friday.
Until now, only major governments have launched a vessel to the space station.
NASA has stressed that SpaceX was a demonstration flight, one of 12 planned SpaceX flights designed to establish whether SpaceX can deliver cargo to the space station, CNN reported.
Currently, NASA has to rely on the Russian space agency to ferry U.S. astronauts to orbit.
Meanwhile, ABC news reported that the rocket carried not only the ashes Mercury astronaut Gordon Cooper, but also of actor James Doohan, who played Scotty on the original "Star Trek" series and the remains of 306 other people.
For the most part, their families paid space services company Celestis between $1,000 and $13,000 to organize sending their ashes into orbit.
According to ABC, the family of Brady Watson Kane, who died last year in a skydiving accident in Colorado, remembers when he told a teacher: "Some day I will take my family to space!" His parents said they will watch the launch on Saturday at Cape Canaveral in Florida.
The ashes are in a canister stored in the second stage of the Falcon 9 rocket, designed to separate from the Dragon capsule nine minutes into flight and orbit the Earth for up to a year before it re-enters the atmosphere and burns up.