NASA astronaut Sunita Williams, Russian cosmonaut Yury Malenchenko and Japan's Akihito Hoshide are set to travel two days before reaching their three colleagues already at the permanent space outpost.
Families and colleagues watched the launch from an observation platform in the Russian-leased cosmodrome in the dry southern steppes of this sprawling Central Asian nation.
The Soyuz jettisoned three rocket booster stages as it was propelled into orbit, which takes just over nine minutes.
At that stage, a doll given to Malenchenko as a mascot by his daughter and suspended over the three astronauts floated out of view on television footage, indicating the craft had escaped the earth's gravitational pull.
Williams gave a thumbs-up sign and waved to onboard cameras as Russian space agency chief Vladimir Popovkin congratulated the crew over radio control.
Malenchenko, who is piloting the Soyuz, is one of Russia's most experienced astronauts and is making his fifth voyage into space.
Williams, who was born in Euclid, Ohio, and raised in Massachusetts, is on her second mission and will further extend the record for the longest sojourn in space for a female astronaut. She spent 195 days at the space station in 2006-2007.
Russians Gennady Padalka and Sergei Revin and U.S. astronaut Joseph Acaba have been working at the space station since mid-May.
The space station, which orbits up to 410 kilometers (255 miles) above the earth, is braced to handle an unprecedented level of traffic.
Japan's HTV3 cargo ship will dock with the space station next week and will be the first of nine craft making contact with the orbiting satellite over a 17-day span.
The Soyuz is schedule to dock Tuesday with the space station at 08:52 a.m. Moscow time (0452 GMT).