The rocket took off yesterday in windy conditions from Russia's space base in Kazakhstan at 2126 GMT.
The trio comprise Russians Oleg Skriprochka and Alexey Ovchinin, and US grandfather of three, Jeff Williams, a veteran of long-duration space missions.
"The Soyuz rocket took off successfully," the Russian space agency Roskosmos said in a statement, adding that the crewship was scheduled to dock with the orbital outpost at 0311 GMT today.
By the end of his half-year trip aboard the ISS, Williams "will become the American with the most cumulative days in space -- 534," NASA says.
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The previous US record was set by astronaut Scott Kelly earlier this year.
Kelly, 52, spent nearly a year in space and returned to Earth earlier this month with a total of 520 days in space, as part of an experiment to test the effects of long-duration spaceflight on the body and mind ahead of a potential future mission to Mars.
Williams has so far notched up over 362 days in space, including three spacewalks.
His previous missions were flown aboard the space shuttle Atlantis in 2000, as well as a trip in 2006 when the station was far smaller -- with only two modules and three crew members.
In 2009 and 2010, Williams served as a flight engineer for three months and later commanded the ISS for the remainder of his half-year mission.
The ISS has been orbiting the Earth at roughly 28,000 kilometres per hour since 1998 and has been continuously occupied since the first expedition in November 2000.