In a pivotal moment for private spaceflight, a towering white rocket lifted a cone-shaped capsule into space early Tuesday on a mission to the International Space Station.

SpaceX's Falcon 9 rocket carried the unmanned Dragon capsule into space after a 3:44 a.m. EDT launch from Cape Canaveral, Fla., marking the first time a private company has sent a spacecraft to the space station.

The mission is considered the first test of NASA's plan to outsource space missions to privately funded companies now that its fleet of space shuttles is retired. SpaceX aims to prove to NASA that its Falcon 9 rocket and Dragon capsule are ready to take on the task of hauling cargo -- and eventually astronauts -- for the space agency.

INTERACTIVE: SpaceX's demonstration mission

"Today marks the beginning of a new era in exploration; a private company has launched a spacecraft to the International Space Station that will attempt to dock there for the first time," said NASA Administrator Charles Bolden in a speech at the cape. "And while there is a lot of work ahead to successfully complete this mission, we are certainly off to good start."

In a separate news conference, Elon Musk,SpaceX's 40-year-old billionaire founder and chief executive, spoke at company headquarters in Hawthorne. It was there that SpaceX employees had gathered, watched and cheered as the Falcon 9 climbed toward the heavens.

"There's so much hope riding on that rocket," he said. "When it worked ... and they saw their handiwork in space and operating as it should, there was tremendous elation. For us, it's like winning the Super Bowl."

But the launch is just the beginning, and the toughest tasks in the mission lie ahead.

The Dragon capsule will rendezvous with the space station as it circles the Earth at about 17,000 mph. Once the Dragon catches up to the station, it must complete a series of complicated tests to determine if it is ready to dock.

If all goes well, the crew aboard the station will snag the spacecraft with a robotic arm and lead it in. SpaceX hopes to dock the Dragon, which is designed to carry up to seven astronauts, as early as Friday.

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NASA hails SpaceX launch as 'a new era' for spaceflight