The rocket took off into the night sky on schedule at 3:11 am (local time), creating a spectacular yellow light. Russian space officials said the launch had gone according to plan.
Their Soyuz TMA-M spacecraft is a modernised version of the ship used by Russia to put humans into the space. It is the first such craft to be fully equipped with a digital measuring system and today's launch was the first time the new design has been used.
The spacecraft is due to dock with the International Space Station (ISS) at 4:02 am (local time) on Sunday.
The crew includes one of Russia's most experience spacemen, Alexander Kaleri, whose first mission to space was in March 1992 just after the fall of the Soviet Union to the now defunct Russian space station Mir.
Kaleri has already made four space flights, logging up 610 days in space and almost 24 hours of spacewalks, his last trip an October 2003 mission to the International Space Station.
Joining him are American Scott Kelly, who has made two spaceflights and Oleg Skripochka, who is making his first space flight.