The rocket carrying the Soyuz TMA-19M spaceship lifted off at 1703 p.m. local time (1103 GMT), beginning its six-hour journey to the International Space Station, and successfully reached its designated orbit about nine minutes later.
As well as Peake, the crew includes commander Yuri Malenchenko, a former Russian Air Force pilot and a veteran of long-duration space flights, and NASA astronaut Tim Kopra.
Peake, 43, a former army major who is on a six-month mission for the European Space Agency (ESA), became the first Briton to go into space since Helen Sharman travelled on a Soviet spacecraft for eight days in 1991.
He is also the first astronaut officially representing the British government and wearing a Union Jack flag on his arm.
The same trio of Malenchenko, Kopra and Peake are set to return to Earth on June 5 next year.