NASA's Shane Kimbrough and Andrei Borisenko and Sergei Ryzhikov of Roscosmos blasted off at 0805 GMT (1335 IST) from the Baikonur cosmodrome in Kazakhstan after their launch had been delayed by nearly one month because of technical issues.
The trio's mission is expected to last just over four months with docking at the orbital laboratory taking place on Friday.
They will join Roscosmos' Anatoly Ivanishin, NASA's Kate Rubins, and Takuya Onishi of the Japan Aerospace Exploration Agency (JAXA), whose return to Earth is set for October 30.
The local Church official who handed over the relics last month, Metropolitan Georgy, had called for the astronauts to receive "God's help in this blessed and good work," the Church said in a statement.
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A few minutes after the launch, Roscosmos confirmed that it had been "successful" and that the three astronauts are scheduled to arrive at the space centre at 9:59 GMT (1529 IST) on Friday.
The 49-year-old Kimbrough, who logged less than 16 days in space during a single mission aboard the US space shuttle Endeavour, will command the expedition aboard the orbital laboratory.
Roscosmos flight engineer Borisenko spent 164 days in space for his first mission aboard the ISS in 2011, while Ryzhikov is entering the cosmos for the first time.
Today's launch is the second Soyuz space flight to be delayed this year after issues with software saw the launch of the newly upgraded Soyuz MS-01 postponed by two weeks over the summer.
Technical mishaps have complicated plans to extend the periods during which the ISS is fully staffed with six astronauts.
In addition to the 2014 Antares explosion, a Russian cargo spacecraft burned up before reaching the ISS in May last year.
The Antares rocket that blasted off from Virginia earlier this week is on course to bring 2,300 kilogrammes of food and equipment to the orbital laboratory and is scheduled to arrive by Sunday.