"Capture is confirmed," said NASA at 9:56 am (1356 GMT) ahead of a second-phase operation to attach the cargo-carrying Dragon to the lab later in the day.
"It looks like we got us a Dragon by the tail," said US astronaut Don Pettit, who was operating the Canadian-built robotic arm from the space station as it reached out and hooked on to the SpaceX capsule.
The two spacecraft were traveling about 250 miles (402 kilometers) above northwest Australia at the time of the grab, NASA said.
A formal berthing of the Dragon later Friday will bring the capsule closer to latch on at the station's Harmony module so its cargo can be unloaded over the coming days, Space X said.
SpaceX's supply ship has now reached the climax of its mission to become the first privately owned craft to berth with the space station, restoring US access to the space outpost after the shuttle program's end.
Only Russia, Japan and Europe currently have supply ships that can reach the ISS. The United States lost that capacity when it retired its space shuttle fleet in 2011.
The capsule blasted off atop the Falcon 9 rocket from Cape Canaveral in Florida on Tuesday.
"Once again Space X has done it. They have just become the first private company to successfully launch their own spacecraft and get captured by the International Space Station robotic arm," said a NASA commentator.
Here are some key facts about the company, known as SpaceX, and its mission.
SPACEX
SpaceX was founded in 2002 by billionaire Internet entrepreneur Elon Musk, the co-founder of PayPal. Musk is also currently the chief executive officer of Tesla Motors, which builds and sells electric cars.
The Hawthorne, California-based company's mission is "to revolutionize space transportation in order to eventually make it possible for people to live on other planets."
The company aims to be able to send people to space aboard its Dragon craft by 2015.
SpaceX employs more than 1,700 people, including a number of former NASA astronauts.
Launch facilities are at the Cape Canaveral Air Force Station and Vandenberg Air Force Base; rocket development facility in McGregor, Texas; and offices in Chantilly, Virginia and the US capital, Washington.
ROCKET
The two-stage Falcon 9 rocket stands at a height of 48.1 meters (158 feet) with the Dragon space capsule on top, and is capable of producing one million pounds of thrust in a vacuum.
All structures, engines, avionics and ground systems are designed, manufactured and tested in the United States.
It is named after the Millennium Falcon, the personal spaceship of the Star Wars characters Han Solo and Chewbacca.
The rocket is powered by nine Merlin engines in the first stage and one in the second stage.
Falcon 9 is powered by liquid oxygen and rocket grade kerosene.
Its first successful launch was on June 4, 2010, followed by a second on December 8, 2010.
SPACECRAFT
Dragon is a reusable spacecraft that was built to carry and return both astronauts and cargo to the International Space Station.
The white capsule stands 4.4 meters (14.4 feet) high and is 3.66 meters (12 feet) in diameter. With its two solar array wings extended, the span is 16.5 meters (54 feet) wide.
Dragon can carry over 3,310 kilograms (7,297 pounds) split between pressurized cargo in the capsule and unpressurized cargo in the trunk.
On this mission, it carries 521 kilograms (1,148 pounds) of cargo for the space lab and will also aim to return a 660-kilogram (1,455-pound) load to Earth.
Dragon is also built to carry up to seven astronauts to the ISS on future missions.
The capsule is maneuvered by 18 Draco thrusters powered by nitrogen tetroxide/monomethylhydrazine propellants.
It is protected by the most powerful heat shield in the world, designed in cooperation with NASA and made of a material called PICA-X.
In December 2010, it became the first private spacecraft to reach orbit and back.
ISS MISSION
May 22: Falcon 9 rocket carrying unmanned Dragon capsule launches from Cape Canaveral Air Force Station at 3:55 am (0755 GMT).
May 24: A flyby of the International Space Station is successfully completed.
May 25: Dragon's landmark rendezvous and berthing to the ISS.
May 26: Live coverage is scheduled of the hatch opening and entry of the Dragon spacecraft. Cargo to be unloaded and capsule restocked with gear to return to Earth.
May 31: The ISS crew will detach Dragon from the space station, and the capsule will perform a series of engine burns that will move it away from the orbiting lab.
About five hours later, the Dragon should reenter the Earth's atmosphere and splash down in the Pacific Ocean, about 450 kilometers (250 miles) off the West Coast of the United States.