A few minutes after their rocket lifted off from the Baikonur Cosmodrome, Russian space agency Roscomos announced that the capsule was "successfully launched into orbit".
NASA administrator Jim Bridenstine confirmed on Twitter that the crew were "safely in orbit" and thanked the US and Russian teams "for their dedication to making this launch a success".
It was the first manned launch for the Soviet-era Soyuz since October 11, when a rocket carrying Russia's Aleksey Ovchinin and US astronaut Nick Hague failed just minutes after blast-off, forcing the pair to make an emergency landing.
They escaped unharmed but the failed launch -- the first such incident in Russia's post-Soviet history -- raised concerns about the state of the Soyuz programme.
The Soyuz is the only means of reaching the ISS since the United States retired the space shuttle in 2011.
Kononenko, McClain and Saint-Jacques showed no signs of worry as they boarded a bus to take them to the launch. They smiled and waved, with Saint-Jacques blowing kisses and giving the thumbs-up to a crowd of well-wishers.
Risk 'part of our profession'
"Risk is part of our profession," the 54-year-old said. "We are psychologically and technically prepared for blast-off and any situation which, God forbid, may occur on board."
McClain, a 39-year-old former military pilot, said the crew looked forward to going up.
"We feel very ready for it," she said.
Saint-Jacques, 48, described the Soyuz spacecraft as "incredibly safe".
The accident highlighted the "smart design of the Soyuz and the incredible work that the search and rescue people here on the ground are ready to do every launch," he said.
In a successful rehearsal for Monday's flight, a Soyuz cargo vessel took off on November 16 from Baikonur and delivered several tonnes of food, fuel and supplies to the ISS.
Russia said last month the October launch had failed because of a sensor that was damaged during assembly at the Baikonur cosmodrome but insisted the spacecraft remained reliable.
Ahead of Monday's launch a Russian Orthodox priest blessed the spaceship on its launchpad, in accordance with tradition.
Of the trio set to reach the ISS six hours after blast-off, both Saint-Jacques and McClain are flying for the first time.
Kononenko is beginning his fourth mission to add to an impressive 533 days in space.
The veteran said the crew would conduct a spacewalk on December 11 as part of an investigation into a mysterious hole that has caused an air leak on the ISS.
Saint-Jacques will be the first Canadian astronaut to visit the space station since Chris Hadfield, who recorded a version of David Bowie's "Space Oddity" on board in 2013.
Saint-Jacques joked that he had received so much training ahead of the flight "that I felt at the end that I could build a Soyuz in my backyard."
'Spacewalking like rugby'
McClain served in Iraq and has represented the United States in women's rugby.
She has said that training to spacewalk resembled the sport since it demands "grit, toughness, mental focus, and more".
Russia-US cooperation in space has remained one of the few areas not affected by a crisis in ties between the former Cold War enemies.
But comments by the combative chief of the Russian space agency, Dmitry Rogozin, have increasingly raised eyebrows.
He recently joked Russia would send a mission to the Moon to "verify" whether or not NASA lunar landings ever took place.
In recent years Russia's debt-laden space industry has suffered a number of mishaps including the loss of cargo spacecraft and satellites.
At a press conference on the eve of the launch, crew commander Kononenko said the astronauts "absolutely" trusted teams preparing for the flight.