The astronauts are expected to reach the orbiting lab on Sunday after blasting off from Cape Canaveral on Saturday to replace four astronauts who have been living up there since March.
A NASA astronaut was joined on the pre-dawn liftoff from Kennedy Space Center by flyers from Denmark, Japan and Russia.
It was the first US launch where every spacecraft seat was occupied by a different country - until now, NASA had always included two or three of its own on its SpaceX taxi flights.
A fluke in timing led to the assignments, officials said.
NASA's Jasmin Moghbeli, a Marine pilot serving as commander, said her crew's makeup demonstrates "what we can do when we work together in harmony".
With her on the six-month mission are the European Space Agency's Andreas Mogensen, Japan's Satoshi Furukawa and Russia's Konstantin Borisov.
"To explore space, we need to do it together," the European Space Agency's director general Josef Aschbacher said minutes before liftoff.
"Space is really global, and international co-operation is key."
The astronauts' paths to space couldn't be more different.
Moghbeli's parents fled Iran during the 1979 revolution.
Born in Germany and raised on New York's Long Island, she joined the Marines and flew attack helicopters in Afghanistan.
The first-time space traveller hopes to show Iranian girls that they, too, can aim high.
"Belief in yourself is something really powerful," she said before the flight.
Mogensen worked on oil rigs off the West African coast after getting an engineering degree.
He told people puzzled by his job choice that "in the future, we would need drillers in space" - think Bruce Willis's character in the killer asteroid film Armageddon.
He's convinced the rig experience led to his selection as Denmark's first astronaut.
Furukawa spent a decade as a surgeon before making Japan's astronaut cut.
Like Mogensen, he's visited the station before.
Borisov, a space rookie, turned to engineering after studying business.
He runs a free-diving school in Moscow and judges the sport, in which divers shun oxygen tanks and hold their breath underwater.
One of the perks of an international crew, they noted, is the food.
Among the delicacies soaring are Persian herbed stew, Danish chocolate and Japanese mackerel.
Lift-off was delayed a day because of extra data reviews for the capsule's life-support system.
Another NASA astronaut will launch to the station from Kazakhstan in mid-September under a barter agreement, along with two Russians.
SpaceX has now launched eight crews for NASA.
Boeing was hired at the same time almost a decade ago but has yet to fly astronauts.
Its crew capsule is grounded until 2024 by parachute and other issues.
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