Stakes were high for Monday's launch, the first to carry a crew since a previous mission to the space station in October was aborted two minutes into the flight after a booster failed to properly separate from the rocket.
This was the first time people rode on one of the Russian rockets since October 11, when two and a half minutes after takeoff, one of the Soyuz MS-10's four boosters broke off, sending the aircraft spinning out of control.
The crew launched at 6:31 a.m. EST Monday from the Baikonur Cosmodrome in Kazakhstan.
The spacecraft has now docked at the space station following four orbits around the earth, NASA says.
Aboard the International Space Station, he will conduct a number of science experiments, with some focusing on the physical effects of the weak gravity astronauts experience in orbit as well as how to provide remote medical care.
NASA and Roscosmos said all onboard systems operated normally and the astronauts felt fine during the six-hour trip the space station.
Last month, Roscosmos tweeted a video of the October 11 failed Soyuz rocket launch that forced NASA astronaut Nick Hague and Russian cosmonaut Alexey Ovchinin into making a harrowing emergency landing.
The trip to space - which had originally been scheduled for December 20 - had to be moved up to ensure that the ISS wouldn't be abandoned for the first time since crew arrived there on November 2, 2000.More news: Gareth Bale agent refuses to rule out Serie A switch
More news: Captain Marvel Poster Heralds the New Trailer
More news: 'We Waited for You': Political Cartoonist Honors George H.W. Bush
Russian Federation said last month the October launch had failed because of a sensor damaged during assembly at the Baikonur cosmodrome, but insisted the spacecraft remained reliable.
European Space Agency astronaut Alexander Gerst, NASA's Serena Aunon-Chancellor and Sergei Prokopyev of Roscosmos will greet the trio when they arrive at the ISS.
Three astronauts were on board Monday for the launch.
Saint-Jacques will be joined by Russian cosmonaut Oleg Kononenko and American astronaut Anne McClain on the mission, with liftoff scheduled for 6:31 a.m. ET Monday if all goes as planned.
Russian space officials took measures to prevent the repeat of such a rocket failure.
"We have confirmation of the spacecraft separation; Soyuz capsule and crew safely in orbit", NASA TV said online in its blow-by-blow commentary of the take-off.
The Soyuz is the only means of reaching the ISS since the U.S. retired the space shuttle in 2011.
Nasa administrator, Jim Bridenstine confirmed on Twitter that the crew were "safely in orbit" and thanked the USA and Russian teams "for their dedication to making this launch a success".