Dubbed Axiom Mission 3 (Ax-3), it is the company's third launch to the space laboratory and the first where all three of the paid seats were bought by national agencies, rather than wealthy individuals.
The spacecraft docked at the ISS at 1043 GMT and boarded within about two hours, according to NASA's livestream of the event.
The SpaceX Crew Dragon fixed to the top of a Falcon 9 rocket blasted off from NASA's Kennedy Space Center in the US state of Florida on Thursday.
It arrived at the ISS, which flies around 260 miles (420 kilometers) above the Earth, after a journey of about 36 hours, according to Axiom Space's website.
Turkish pilot and air force colonel Alper Gezeravci is joined by Walter Villadei, an Italian air force colonel who has previously flown to the edge of space on a Virgin Galactic space plane, and by Marcus Wandt from Sweden, who is representing the European Space Agency.
They are led by Axiom's Chief Astronaut Michael Lopez-Alegria, a Spanish and US citizen and former NASA astronaut.
The Axiom-3 team were welcomed with hugs by the seven crew members already aboard the ISS -- from Japan, Denmark, the United States and Russia.
- 'Pretty exciting' -
"We have doubled the number of nationalities on board the space station, going from four to eight, which I think is a great testament to the international collaboration which underpins this marvelous space station," said ISS commander Andreas Mogensen in a livestream, welcoming the four-member Axiom crew.
"The ride uphill was pretty exciting. It never gets old," added Axiom commander Lopez-Alegria.
"I think we probably spent a few more hours in Dragon than we felt like we needed to," he said, smiling. "But it was all good."
The new arrivals will spend about two weeks carrying out 30 experiments, learning more about the impact of microgravity on the human body, advancing industrial processes and more.
Axiom Space was founded in 2016 by Michael Suffredini, a former ISS program manager for NASA, and entrepreneur Kam Ghaffarian.
In addition to organizing private missions to the orbital outpost, the company is developing spacesuits for future NASA missions to the Moon.
It is also building a commercial space station that it intends to initially attach to the ISS, then separate and orbit independently sometime before the ISS is retired.
The exact costs of the Ax-3 have not been disclosed, but in 2018 when the company first announced the program, which involves chartering SpaceX hardware and paying NASA for services, it set a price tag of $55 million per seat.
More recently, Hungary was reported by spacenews.com to be planning a $100 million deal with Axiom for a future mission involving one astronaut.
Britain, which is striving to build a post-Brexit space strategy, has also signed an agreement for a future mission carrying UK astronauts.
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