If the six-day demo goes well, SpaceX could launch two astronauts this summer under NASA's commercial crew programme.
Doug Hurley and Bob Behnken, who will be the first human crew on the Dragon, sent commands from their base in California to position the vehicle before it made its final approach to the ISS.
The last time US astronauts were launched from Kennedy Space Center on Florida's Space Coast was in July 2011 aboard the Space Shuttle Atlantis.
SpaceX founder Elon Musk noted the accomplishment in a post-launch news conference with NASA officials Saturday morning.
"Man, we were all working so hard to get to the start of flying this new vehicle-it was super exhilarating", he said. The company has developed a capsule of its own called the Starliner.
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The first and second stages separated without incident, placing Dragon in Earth's orbit 11 minutes after take-off.
NASA published a series of photos showing the arrival of SpaceX's Crew Dragon reusable spacecraft to the International Space Station.
"Early on, our goal was human space flight", he said.
"I really believe in the future of space and I think it's important we become a spacefaring civilisation and be out there among the stars, and I think that's one of the things that makes people excited about the future", he said.
For operational missions, Crew Dragon will be able to launch as many as four crew members and carry more than 220 pounds of cargo, enabling the expansion of the inhabitants of the space station, increasing the time dedicated to research in the unique microgravity environment, and returning more science back to Earth.
SpaceX's Crew Dragon is docked to the International Space Station.
The capsule's lone passenger for launch - a mannequin wearing a white SpaceX spacesuit - remained strapped into its seat as the station's U.S., Canadian and Russian crew removed supplies and photographed the spotless white interior. The Crew Dragon was designed with an emergency escape system that can carry people out to safety if something goes wrong.
Saint-Jacques and Kononenko were the first to enter the Crew Dragon after opening the hatch. "Just one more milestone that gets us ready for our flight coming up here".