The two men, joined by US astronaut Christina Koch, lifted off from Russia's Baikonur cosmodrome in Kazakhstan at the expected time 1914 GMT.
"The Soyuz is now in orbit and the crew are on their way to the International Space Station," said a commentator on NASA television.
The Russian space agency Roscosmos confirmed it had entered orbit.
The lift off was closely watched after the two men's space journey was cut short in October when a technical problem with their Soyuz rocket triggered a launch abort two minutes into the flight.
Both men escaped unharmed.
It was the first such accident in Russia's post-Soviet history and a major setback for its once proud space industry.
Speaking to reporters ahead of their six-month mission, flight commander Ovchinin said that some faulty components in the launch vehicle had been found and replaced this week.
"Yesterday they found some minor malfunctions," the 47-year-old said on Wednesday, March 13.
He insisted that the launch vehicle was in good shape.
"There are no problems," Ovchinin said.
Hague, 43, said he was looking forward to the flight – his second attempt to get into space.
"I'm 100 percent confident in the rocket and the spaceship," he said.
The October abort was caused by a sensor damaged during the rocket's assembly.
'Old but reliable'
Space expert Vadim Lukashevich said last-minute replacements were nothing out of the ordinary.
"The Soyuz is an old but reliable machine," he told AFP.
Russia's space industry has in recent years suffered a lot of mishaps including the loss of cargo spacecraft and numerous satellites.
Ovchinin, who spent 6 months at the ISS during a previous mission in 2016, has been keen to play down the drama of the October emergency landing.
The abort was "a little disappointing" after preparations that lasted a year-and-a-half but also "an interesting and needed experience" that tested the depth of the space program's preparedness, he said.
Koch, Hague and Ovchinin's six-hour flight Thursday will be closely watched for another reason too.
SpaceX's successful test launch to the ISS of its Dragon vehicle has challenged an eight-year monopoly on travel to the space station enjoyed by Russia ever since NASA stopped launches of the Space Shuttle.
Speaking to reporters, the trio and their three-man backup crew stressed cooperation rather than competition following the Dragon mission, seen by some as the dawn of an era of commercial space travel driven by businessmen such as Elon Musk who owns SpaceX.
Koch, a 40-year-old space rookie, called the SpaceX success a "great example of what we've been doing for a very long time."
"And that is cooperating among partners and making things that are very difficult look easy," she said.
There had already been one successful manned launch to the ISS since the failed Soyuz mission.
The trio's arrival will return the orbiting laboratory's crew to 6 in all.
Oleg Kononenko of the Russian space agency Roscosmos, Anne McClain of NASA and David Saint-Jacques of the Canadian Space Agency blasted off to the orbiting outpost in December and are expected to greet their new crewmates early on Friday, March 15.
During their mission McClain, Saint-Jacques, Hague and Koch are set to perform the first spacewalks of their careers.
This week in a lighter moment Hague offered an insight into the specifics of personal grooming aboard the ISS.
"In space, we'll use clippers attached to a vacuum device so that the hair particles don't float around or get stuck in our vent systems," he wrote on Twitter.
The International Space Station – a rare area of cooperation between Moscow and Washington – has been orbiting the Earth at roughly 28,000 kilometres per hour since 1998. – Rappler.com