The trip by the Soyuz spacecraft is scheduled to last two days instead of the usual fast-track six hours since the ISS has raised its orbit to avoid space debris.
The trio launched on schedule at 0437 GMT from the same launchpad in Kazakhstan used by Yuri Gagarin on his historic entry into the cosmos in 1961.
"The crew is doing well, everything is in order onboard," relayed mission control.
They are expected to dock with the ISS on Friday at 07:42 GMT.
Most recent launches including the latest in July have used a six-hour approach to the ISS, orbiting the Earth four times, which is less gruelling for astronauts.
The two-day approach, orbiting Earth 34 times, was the norm until 2013.
Russia's space agency acting chief Alexander Ivanov, quoted by Interfax news agency, said the two-day journey was chosen as "the safest and most reliable."
The ISS on Monday also manoeuvred to raise its orbit again by one kilometre to ensure that the next batch of departing astronauts land in the correct location on Earth on September 12, mission control said.
The latest take-off marked the 500th launch from the Gagarin launchpad named after the Soviet space pioneer.
Volkov will stay on at the ISS for 188 days, while both Aimbetov and Mogensen will return to Earth next week after 10 days.
Mogensen is the first Dane to enter space. "It's a great honour for me to represent Denmark as an astronaut," he said last month.
Brightman, a 55-year-old soprano known for her roles in Andrew Lloyd Webber musicals, pulled out of her plan to fly as a space tourist on the flight for a fee of $52 million, citing family reasons.
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