Monday's launch from Kazakhstan is the first successful manned mission to the space lab since an aborted Soyuz launch in October.
NASA and Roscosmos said all onboard systems were operating normally and the crew was feeling fine.
McClain, Saint-Jacques and Kononenko will spend more than six months doing research and experiments in biology, Earth science, physical sciences and technology.
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.
The families of the crew, other astronauts, and space officials from several nations breathed a sigh of relief after observing the flawless launch, with October's Soyuz rocket failure still on the minds of many. They are scheduled to return to Earth on December 20.
The launch comes less than two months after a booster failure forced a Soyuz spacecraft carrying Russian cosmonaut Alexey Ovchinin and U.S. astronaut Nick Hague to make an emergency landing. They managed to emerge safely despite the harrowing ordeal. At that time, the space station and the spacecraft were 251 miles over the Atlantic Ocean, just east of the Caribbean.
It was the first manned Soyuz rocket to blast off since an aborted October 11 launch when a Soyuz rocket meant to carry Russian cosmonaut Aleksei Ovchinin and USA astronaut Nick Hague to the ISS failed two minutes into its flight.
A Russian Soyuz spacecraft carrying three astronauts, including one American, has successfully docked with the International Space Station.
Neither man was injured in the incident, which was blamed on a faulty sensor damaged during the rocket's assembly. The Russian rocket carries US astronaut Anne McClain, Russian cosmonaut Oleg Kononenkoâ€Ž and CSA astronaut David Saint Jacques.