The incoming crew will spend five months in space before returning to Earth.
Their mission began with a late-night launch from the Russian-leased Baikonur launch pad in Kazakhstan.
It was the first time a space crew has taken such a direct route to the orbiting lab. Cassidy, Vinogradov and Misurkin are the first crew to reach the station after only four orbits instead of the standard 50-hour flight to reach the station.
The new maneuver was tested successfully by three Russian Progress cargo ships, unmanned versions of the Soyuz used to ferry supplies to the space station.
Vinogradov joked at a pre-launch news conference at Baikonur that the journey to the station would be so quick that it could allow the crew to even carry ice cream as a present to the three men currently manning the orbiting outpost.
"It wouldn't melt in such a short time," he said.
On a more serious note, Vinogradov added that the shorter flight path would reduce the crew's fatigue and allow astronauts to be in top shape for the docking. He said that it takes about five hours for the human body to start feeling the impact of zero gravity, so the quicker flight would allow the crew to more easily adapt to weightlessness in much roomier space station interiors.
The downside of the accelerated rendezvous is that the crew will have to stay in their spacesuits, which they don hours before the launch, through the entire approach maneuver.
Other Russian cosmonauts in the past have described the two-day approach maneuver in the cramped Soyuz as one of the most grueling parts of missions to the orbiting station. The spheroid orbiting capsule allows the crew to take off their bulky spacesuits, change into more comfortable clothes and use a toilet, but its interior is extremely confined.
The ship's spartan layout lacks adequate heating and fails to provide an opportunity for the crew to get hot food. It contrasts sharply with the spacious U.S. space shuttle, whose retirement has left Soyuz as the only means to deliver crews to the space outpost.
Russian space officials said the longer approach was necessary at a time when the station was in a lower orbit required for the shuttle flights. After they ended, it was raised from 217 miles to 249 miles, making a quicker rendezvous possible.
NASA is working on the development of its new generation Orion spacecraft. Orion's first trip is an unmanned mission in 2017, and the first manned mission is set for 2021.