McClain, Saint-Jacques and Kononenko will spend more than six months at the space station doing research and experiments in biology, Earth science, physical sciences and technology.
The International Space Station is about to get three new residents.
The manned mission was the spacecraft's first since October's forced emergency landing.
The three-person crew's mission was originally scheduled for later this month, but officials brought it forward to avoid the ISS being left unmanned when its current crew return to earth.
A few minutes after the rocket lifted off the Russian space agency Roscomos announced that the capsule was "successfully launched into orbit".
The launch comes after a Soyuz rocket carrying Russia's Aleksey Ovchinin and United States astronaut Nick Hague failed on October 11 just minutes after blast-off, forcing the pair to make an emergency landing.
The pair escaped unharmed, but the failed launch was the first such incident in Russia's post-Soviet history and a new setback for the country's once proud space industry.
Since NASA retired the space shuttle in 2011, Russian Soyuz rockets have been the only way to get people to the International Space Station.
"I am completely astounded by everything I have seen", Saint-Jacques said during a brief conversation with family members on the ground at the Baikonur Cosmodrome launch site in Kazakhstan. They smiled and waved, with Saint-Jacques blowing kisses and giving the thumbs-up to a crowd of well-wishers.
Crew commander Kononenko, 54, said during a press conference on the eve of the launch that "risk is part of our profession".
Reports say a Russian Orthodox priest blessed their rocket before its flight on Monday, as per tradition. Blastoff kicked off a 6-hour-, four-orbit-long journey to the space station for the trio of astronauts on board. "We are psychologically and technically prepared for blast-off and any situation which, God forbid, may occur onboard". Plus, it's the first time a Canadian astronaut (Saint-Jacques) is headed to space since Chris Hadfield, who lead the International Space Station five years ago and was known for his publicity success.
"Space represents a lot of opportunities for a lot of Canadians", he said at the agency office.