The 21m-long vehicle lifted off at 1810 California time (0110 GMT) and rose to an altitude of 320km (200 miles).
Mr Musk, who co-founded the internet financial system PayPal, wants to lower the cost of access to space.
The flight did not achieve all its goals, but the he said it demonstrated the vision of his Space Exploration Technologies Corporation (SpaceX).
The mission was the second attempt to loft the rocket; the first, in March 2006, ended when a fire fed by a fuel leak led to the shut down of the main-stage engine just 29 seconds after lift-off.
