The Olympic torch has gone into space twice before, in 1996 and 2000, but it has never been taken on a space walk.
"Our goal here is to make it look spectacular," Kotov told reporters before his mission began. "We'd like to showcase our Olympic torch in space ... Millions of people will see it live on TV and they will see the station and see how we work."
While the red-and-silver torch, designed to evoke the feathers of a Firebird from Russian folklore, is taken outside space station 250 miles (400 km) above Earth, the flame will remain lit on the ground below.
As well as replacing the gas flame, Russian engineers have equipped the torch with a tether to keep it from floating out of the cosmonauts grip in weightlessness. "It was reworked ... so that it doesn't fly away," said Sergei Krikalev, head of the Cosmonauts' Training Centre outside Moscow.
As part of its 65,000-km relay, the torch has travelled to the North Pole on an atomic-powered ice breaker.
It will still go to the peak of Europe's highest mountain, Mount Elbrus, and the depths of Siberia's Lake Baikal before reaching Sochi on the Black Sea for the start of the Games on February 7.
"The Olympics are a huge international event that takes many, many countries cooperating and working together to pull off such a tremendous event," Mastracchio, 53, told reporters.
"So in a small way, I think it's great that we bring this symbol up to the international space station, which is another representation of international cooperation."
The torch-bearing trio's arrival at the orbital station, a $100 billion project of 15 nations, will briefly swell its crew to nine people - the most on board the outpost since America's last shuttle mission in 2011.
The torch will be brought back to Earth by Russian Fyodor Yurchikhin, American astronaut Karen Nyberg and European Space Agency astronaut Luca Parmitano on Nov 11.