Axiom Space, the Houston company that arranged the deal, put the ticket price at more than $US65 million ($A100 million) per customer.
SpaceX's Falcon rocket blasted off from NASA's Kennedy Space Center on Wednesday two weeks late because of space station leak concerns.
The capsule on top carried not only the three newcomers to space - none of whom were alive when their countries' first astronauts launched - but America's most experienced astronaut, Peggy Whitson.
Besides Whitson, the crew includes India's Shubhanshu Shukla, a pilot in the Indian Air Force; Hungary's Tibor Kapu, a mechanical engineer; and Poland's Slawosz Uznanski-Wisniewski, a radiation expert and one of the European Space Agency's project astronauts sometimes pressed into temporary duty.
The astronauts are due to arrive at the orbiting lab on Thursday.
As well as dozens of experiments, the astronauts are flying food that celebrates their heritage: Indian curry and rice with mango nectar; spicy Hungarian paprika paste; and freeze-fried Polish pierogies.
Hungary's first astronaut, Bertalan Farkas, cheered on Kapu from the launch site.
"For such a small country as Hungary, it is really important to collaborate in a peaceful international space co-operation," Farkas told The Associated Press.
Farkas launched with the Soviets in 1980, taking a teddy bear in a cosmonaut suit that went back up with Kapu.
India and Poland's original astronauts also launched with the Soviets in the late 1970s and 1980s.
Uznanski-Wisniewski carried up the Polish flag worn on his predecessor's spacesuit, noting that Miroslaw Hermaszewski was his biggest supporter until his death in 2022.
India's first astronaut, Rakesh Sharma, could not make it to Florida for the launch; Shukla said he had been a mentor "at every step of this journey" and was flying a surprise gift for him.
It was Axiom's fourth chartered flight to the space station since 2022 and Whitson's second time flying as an Axiom crew commander and chaperone.
Once opposed to non-traditional station guests, NASA now throws out the welcome mat, charging for their food and upkeep while insisting that an experienced astronaut accompany them.
It's all part of NASA's push to open space - moon included - to private businesses.
Axiom is among several US companies planning to launch their own space stations in the next few years, with the aim for them to be up and running before the international station comes down in 2031 after more than three decades of operation.
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