The Soyuz-FG rocket and Soyuz-TMA capsule, emblazoned with the symbols of the Sochi Games and the Olympic rings, blasted off for the International Space Station (ISS) on time at 8:14 am (0414 GMT), it said.
"The Soyuz FG rocket and the Soyuz-TMA capsule launched successfully from Baikonur cosmodrome at 8:14 am Moscow time," the Russian space agency said in a statement.
The rocket decorated with the mainly blue Olympic logo and the words Sochi 2014 blasted off into a clear skies from the Kazakh steppe.
The astronauts will now take the six-hour trip to the ISS, where they will join six incumbent crew.
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In an unprecedented move, two Russian cosmonauts who are already on board the ISS, Sergei Ryazansky and Oleg Kotov, will take the torch on a space walk from 1430 GMT on Saturday.
The torch will remain unlit at all times for safety reasons.
The torch will then be taken back to Earth by three astronauts now finishing their five-and-a-half-month mission who are due to touch down in Kazakhstan on Monday at 0250 GMT.
In a spectacular torch relay, Russia last month took a lit Olympic torch to the North Pole on a nuclear-powered icebreaker.