The rocket lifted off at 3:02am local time on Wednesday (2102GMT on Tuesday) and is to dock with the orbiting laboratory about six hours later.
The capsule is carrying Oleg Kononenko of Russia, NASA's Kjell Lindgren and Kimiya Yui of Japan.
They will join Gennady Padalka, Mikhail Kornienko and Scott Kelly.
The latter two are more than four months into a nearly year-long mission on the space station.
Wednesday's launch was postponed by about two months after the April failure of an unmanned Russian cargo ship, which raised concerns about Russian rocketry.
Another Russian cargo ship was successfully launched in early July.