The test flight of the Dragon space capsule, which launched atop SpaceX's Falcon 9 rocket, aims to show that commercial industry can restore US access to the ISS after NASA retired its space shuttle fleet last year.
The mission is set to include a fly-by and berthing with the station in the next three days, before returning to Earth at the end of this month.
Hugs all round
Moments after liftoff, the cargo-carrying spacecraft entered orbit as planned, as video images showed mission control staff at SpaceX headquarters in Hawthorne, California jumping from their seats, hugging and clapping.
"Falcon flew perfectly!! Dragon in orbit," SpaceX founder and Internet entrepreneur Elon Musk wrote on the micro-blogging website Twitter. "Feels like a giant weight just came off my back."
The apparently flawless launch came after an attempt on Saturday was scrubbed at the last second when computers detected high pressure in the central engine of the Falcon 9.
SpaceX engineers discovered the root cause was a faulty check valve and repaired it the same day.
Where to from here?
No humans are travelling aboard the Dragon, but six astronauts are already at the $100-billion space lab to help the capsule latch on, to unload supplies and then restock the capsule with cargo to take back to Earth.
On Thursday the spacecraft's sensors and flight systems are to undergo a series of tests to see if the craft is ready to berth with the space station, including a complicated fly-under at a distance of about 2.5km.
If NASA gives the green light, the Dragon will then approach the ISS on Friday in an attempt to berth with the station.
The astronauts on board the ISS will maneuver the station's robotic arm to help capture the capsule and attach it to the orbiting research outpost.
The hatch of the Dragon is set to open on Saturday for unloading 521kg of cargo for the space lab and restocking it with a 660kg load to return to Earth.
On May 31, the Dragon is to detach from the station and make a safe landing in the Pacific Ocean off the coast of southern California.
All aboard in race to space
California-based SpaceX, owned by billionaire Musk, who also co-founded PayPal, is the first of several US competitors to try sending spacecraft to the ISS with the goal of restoring US access to space for human travellers by 2015.
The company successfully test-launched its Falcon 9 rocket in June 2010, then made history with its Dragon launch in December of that year, becoming the first commercial outfit to send a spacecraft into orbit and back.
Its reusable Dragon capsule has been built to carry both cargo and up to seven crew members.
Until now, only the space agencies of Russia, Japan and Europe have been able to send supply ships to the ISS.
The three-decade US shuttle program, which ferried astronauts and cargo to the research outpost, ended for good in 2011, leaving Russia as the sole taxi to the ISS until private industry comes up with a replacement.
What's the price tag?
The US space agency has given SpaceX about $390 million so far of the total $680 million that the company has spent on cargo development. SpaceX has a $1.6 billion contract with NASA for future supply missions.
Orbital Sciences Corporation is working on its own supply ship, with a preliminary launch scheduled for later this year, and has a $1.9 billion contract with NASA.
SpaceX also gets funding from NASA for a separate effort to develop a commercial crew vehicle for carrying astronauts to space, along with competitors Blue Origin, Boeing and Sierra Nevada.
In a few years' time, SpaceX says it will be able to undercut the hefty price NASA pays Russia for US astronauts to get a seat aboard the Soyuz space capsule - about $63 million a ticket.