Space Exploration Technologies Corp. successfully blasted its first commercial satellite into space on Tuesday, establishing itself as a low-cost alternative to legacy satellite-launch providers backed by U.S. and foreign governments.

After three aborted countdowns in the past week—including one halted seconds before lift-off—the launch from Florida's Cape Canaveral Air Force Station went without a hitch. The upgraded Falcon 9 rocket placed a three-ton communications satellite owned by SES S.A. into high-earth orbit. About...