The mammoth vehicle - the most powerful since the shuttle system - lifted clear of its pad without incident to soar high over the Atlantic Ocean.
It was billed as a risky test flight in advance of the lift-off.
The SpaceX CEO said the challenges of developing the new rocket meant the chances of a successful first outing might be only 50-50.
"I had this image of just a giant explosion on the pad, a wheel bouncing down the road. But fortunately that's not what happened," he told reporters after the event.
With this debut, the Falcon Heavy becomes the most capable launch vehicle available.
It is designed to deliver a maximum payload to low-Earth orbit of 64 tonnes - the equivalent of putting five London double-decker buses in space.
Such performance is slightly more than double that of the world's next most powerful rocket, the Delta IV Heavy - but at one third of the cost, says Mr Musk.For this experimental and uncertain mission, however, he decided on a much smaller and whimsical payload - his old cherry-red Tesla sports car.
A space-suited mannequin was strapped in the driver's seat, and the radio set to play a David Bowie soundtrack on a loop.
The Tesla and its passenger have been despatched into an elliptical orbit around the Sun that reaches out as far as the Planet Mars.
The Falcon Heavy is essentially three of SpaceX's workhorse Falcon 9 vehicles strapped together. And, as is the usual practice for SpaceX, all three boost stages - the lower segments of the rocket - returned to Earth to attempt controlled landings.