The unmanned Dragon capsule was launched on 2 February on its first test flight.
Only an instrumented dummy was onboard the capsule while it docked at the ISS, where three astronauts are currently present.
If the six-day demonstration goes well, SpaceX could launch two astronauts this summer under Nasa’s commercial crew programme.
Both astronauts – Doug Hurley and Bob Behnken – were at SpaceX mission control in southern California, observing all the action.
“Just super-excited to see it,” Mr Behnken said minutes after the link-up. “Just one more milestone that gets us ready for our flight coming up here.”
SpaceX, the company founded by Elon Musk, had previously sent cargo capsules to the space station, but a spacecraft designed for crew is a new milestone.
The Dragon capsule docked autonomously at the ISS instead of relying on the station’s robot arm.
Mr Behnken said that is the way it should work when he and Mr Hurley are on board but they may push a button or two and will have the ability to intervene, if necessary.
The station’s astronauts also sent commands for the Dragon to retreat and then move forward again, before the capsule closed in for good.
SpaceX employees at company headquarters in Hawthorne, California, cheered and applauded as the Dragon pulled up and docked at the orbiting lab, nearly 260 miles above the Pacific, north of New Zealand.
They burst into applause again, several minutes later, when the Dragon’s latches were tightly secured.
The capsule’s lone passenger – a mannequin wearing a white SpaceX spacesuit – also was going to be welcomed on board.
The test dummy – or Smarty as SpaceX likes to call it, given all the instrumentation – is named Ripley after the lead character in the science-fiction Alien films.
Dragon will remain at the space station until 8 February, when it undocks and aims for a splashdown in the Atlantic, a couple of hundred miles off the Florida coast.
Like Ripley, the capsule is rigged with sensors to measure noise, vibration and stresses, and to monitor the life-support, propulsion and other critical systems throughout the flight.
Additional reporting from agencies
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