Less than six hours after take-off from Russia's Baikonur cosmodrome, Kjell Lindgren from the US, Kimiya Yui of Japan and Russian Oleg Kononenko safely arrived at the orbital outpost.
The flight had been postponed after the April launch of a cargo rocket failed.
Manned flights to the ISS are currently only possible with Russia's ageing Soviet space technology.
The US retired its Space Shuttle operation in 2011.
Thursday's mission capsule connected to the International Space Station about 250 miles (400km) above Earth at 01:45 GMT.
The three astronauts had been set to take off in May but Moscow was forced to delay the flight after the 28 April crash when an unmanned Soyuz cargo rocket had failed to reach the station and burned up in the atmosphere before crashing back to Earth.
"It's certainly no fun to see several of the cargo vehicles undergo mishaps," Mr Lindgren said. "It underscores the difficulty of this industry and how unforgiving the space environment," he told a news conference ahead of the launch.
For both the US astronaut and for Kimiya Yui, it is their first time in orbit.
The Japanese astronaut said he was taking some sushi along as a treat for the others.
The team has joined the existing ISS crew of Russians Gennady Padalka, Mikhail Kornienko and Scott Kelly from the US.
Aside from Russia's Soyuz rockets that largely date back to Soviet technology, two privately owned US companies flying cargo the ISS have also lost rockets in recent launch failures.
Both SpaceX and Orbital ATK currently remain grounded following accidents last month and in October last year.
Astronauts from Russia, the US and Japan have successfully docked at the International Space Station.