"On Thursday at 03:15 (0015 GMT) a Soyuz-2.1B rocket was successfully launched carrying a satellite for the Russian military," the Russian defence ministry said in a statement.
The satellite reached its orbit at the set time, according to the ministry.
"This is the first launch of a rocket from the Soyuz family since the October 11 accident," Russia's space agency chief Dmitry Rogozin wrote on Twitter.
It was the third launch of a Soyuz rocket from Russia's northern Plesetsk launch pad this year, the military said.
The satellite launch had originally been planned for October 19 but was postponed after the accident that saw two astronauts make an emergency landing minutes after blast-off from Russia's Baikonur cosmodrome.
The failed launch involved an older Soyuz-FG rocket, which is "in principle the same rocket" as used in Thursday's launch but has less engine power in the third stage, space expert Konstantin Kreidenko told AFP,
Russia will switch to the Soyuz-2.1 series for launches to the ISS by 2020, replacing the Soyuz-FG, which has been in use for 20 years, Kreidenko said.
Russia's Roscosmos space agency has set up a commission to investigate the failed launch to the ISS and is set to announce its findings on October 30.
Russia has in the meanwhile suspended all Soyuz launches but this does not apply to launches by the military such as Thursday's satellite launch.