The California-based company, Astra Space, was successful in reaching the correct altitude.
The startup, which specializes in launch vehicles, blasted its rocket from the Pacific Spaceport Complex in Alaska on Saturday.
The 43-foot rocket, which was carrying a test payload for the US Space Force, achieved its target altitude after nine minutes.
Musk and other figures in the space industry praised the achievement. "Good luck!" The CEO of the two companies wrote in a message to the other person that it was not easy.
According to CNBC, Astra is one of a few US space companies that have reached space with a privately funded rocket.
Chris Kemp said on the company's website that it was difficult to do.
The team has worked so hard for so many years and everyone is incredibly passionate.
The company's launch follows a number of failed attempts. In August, one of the main engines on the vehicle shut down just after liftoff, Insider reported.
According to CNBC, the company hopes to launch one small rocket a day by the year 2025.
Musk said that his company will fly its spaceship into space in early 2022. He warned that the initial attempt might fail.
In December, the company will conduct tests ahead of the flight in January.
