The space shuttle might be history, but America's pursuit of the stars is still flying high.

Private rocket and spacecraft design firm SpaceX, headed by Tesla Motors (NASDAQ:TSLA) CEO Elon Musk, made history on Sunday evening by launching the first official private mission to resupply  the International Space Station. The company's Dragon spacecraft lifted off from Cape Canaveral in a launch Musk called "an unqualified success." It's the first of 12 planned cargo hauls to the station for SpaceX as part of a $1.6 billion contract awarded by NASA. SpaceX had previously made a launch to the station back in the spring, but that was a test flight  for the Dragon capsule.

If all goes according to plan, the capsule will dock with the space station on Oct. 10 and stay for more than two weeks before headed back to Earth  and a splashdown in the sea. Unlike supply capsules  from other major players in space, such as Japan and Russia, SpaceX designed the Dragon craft as a reusable module – inspiring hope at NASA of a new, privately funded era of American space exploration in the post-shuttle period.

SpaceX is the leader among several players in the private space-launch business, its notable competitor being Virginia-based Orbital Sciences (NYSE:OA). The latter has also made strong headwinds recently, with the first stage  of the company's Antares rocket rolling out to the launchpad on Oct. 1. If tests prove well, Orbital plans to send its own Cygnus capsule to the space station in the near future – providing NASA with several options in the burgeoning space industry. The agency's already handed Orbital a $1.9 billion contract for eight unmanned cargo runs, providing SpaceX with a legitimate rival.

For now, SpaceX will enthusiastically embrace its nascent success. SpaceX President Gwynne Shotwell summed up the excitement  at the young company, telling the Associated Press: "[The Dragon is] driving its way to station, so that's just awesome."