Russia's space agency Roscosmos has now successfully launched five Soyuz rockets since the incident, and does not believe there is a chance of the failure repeating.
The journey to the orbital lab takes six hours, with docking expected at 5.36pm GMT.
The previous launch - involving NASA's Nick Hague and Russia's Aleksey Ovchinin - had to be aborted after the first stage of the rocket failed to separate as planned.
Kononenko, McClain and Saint-Jacques were all smiles as footage broadcast by NASA TV showed them preparing to enter the Soyuz capsule before launch.
"Risk is part of our profession", the 54-year-old said at a press conference.
RFE also quoted McClain, 39, saying: "We feel very ready for it".
Spokane astronaut Anne McClain blasts off to International Space Station
The Soyuz spacecraft is now the only vehicle that can ferry crews to the space station, but Russian Federation stands to lose that monopoly in the coming years with the arrival of SpaceX's Dragon and Boeing's Starliner crew capsules.
Afterward, investigators said they believed other Soyuz models may have been defective, but said additional checks had been introduced.
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.
Roscosmos, the Russian space agency, is now the only organisation transporting astronauts to the ISS after Nasa ended its space shuttle flights in 2011.
The three Expedition 58 astronauts will briefly join NASA astronaut Serena Auñón-Chancellor, Alexander Gerst of the European Space Agency and cosmonaut Sergey Prokopyev on the ISS.
During their mission, members of the crew are scheduled to embark on a spacewalk to further probe a mysterious hole that caused a loss of air pressure on-board the ISS in August.
The experiment could pave the way to new treatments for muscular conditions for people on Earth, according to the UK Space Agency.