It saw SpaceX blow up the Falcon 9 rocket carrying the Crew Dragon astronaut capsule, which then drifted safely back down to Earth.
Mr Musk said the successful test had paved the way for real launches in SpaceX's second quarter, which would take it to June at the very latest.
"We're highly confident that the hardware will be ready in Q1, most likely end of February but no later than March," Musk said. "And we think it appears probable that the first crewed launch would occur in the second quarter."
That crewed launch would be the final test, allowing Nasa's commercial crew programme to become fully operational.
A successful mission carrying astronauts would be a major breakthrough for the US, which is currently unable to send people to the International Space Station from its own soil, and instead must rely on the Russian space agency.
Rather than building its own spacecraft for the missions, Nasa has funded Boeing and SpaceX as they attempt to build their own capsule systems and rockets. If that plan is successful, it will allow for the US to launch its own astronauts for the first time since the end of Nasa's space shuttle programme, in 2011.
In the abort test, Crew Dragon detached from the Falcon 9 rocket at "more than double the speed of sound," Musk told reporters, at 131,000 feet (40km) above the Atlantic Ocean – roughly twice the altitude of a commercial jetliner.
"It is a picture perfect mission. It went as well as one can possibly expect," Musk said at a press conference.
National Aeronautics and Space Administration administrator Jim Bridenstine also described the test as a success.
Crew Dragon, an acorn-shaped pod that can seat seven astronauts, fired on-board thrusters to detach itself from the rocket less than two minutes after liftoff, simulating an emergency abort scenario to prove it can return astronauts to safety. Each stage of the test prompted loud cheers from SpaceX crew members watching the footage from back on land.
The capsule deployed four parachutes to slow its descent, and carried two human-shaped test dummies on seats fitted with motion sensors to collect data on the immense g-force – the effect of acceleration on the body – astronauts would be subjected to during abort.
Additional reporting by agencies