Astronauts onboard the platform used its robotic arm to grasp the vehicle and attach it to a berthing port.
Dragon is the first commercial vessel to visit the space station.
It is also the first American ship to go to the orbiting laboratory since NASA retired its shuttles last year.
US astronaut Don Petit was inside the ISS at the controls of the Canadarm2.
Don Petit reached out with the robotic appendage and grabbed the Dragon capsule at 13:56 GMT.
“Houston, looks like we got us a Dragon by the tail,” Don Petit radioed to NASA mission control in Texas.
The capture was met by applause from controllers, at the NASA centre and at the SpaceX HQ in Hawthorne, California, where Dragon’s flight is also being overseen.
Just under two hours later, Don Petit used the arm to position the Dragon under the ISS’s Harmony connecting node. A good seal was confirmed at 16:02 GMT.
The ISS crew will go inside the ship on Saturday to unload its stores.
The attachment of Dragon to the platform marks a significant milestone in the history of human spaceflight.
Traditionally, this field of endeavor has been the preserve of government-owned and operated vehicles.
But NASA is looking to save money that it can then re-invest in some of its other programmes far beyond Earth, at asteroids and at Mars.
It believes this can be achieved by contracting out the more routine tasks in low-Earth orbit to the private sector.
To that end, it is providing seed funding of approximately $800 million to SpaceX and another company, Orbital Sciences Corporation, to help them develop new rocket and cargo vehicles.
Orbital’s rocket is called Antares, and its freighter is known as Cygnus. The pair should go into space together for the first time before the end of the year.
Once these companies have proved the performance of their systems, they will start to receive lucrative ISS re-supply payments.
For SpaceX, its contract is valued at $1.6 billon and calls for a minimum of 12 Dragon cargo missions to the ISS.
But freight is just the start. The ferrying of crews to and from the ISS will be the next service NASA buys in.
SpaceX wants this business as well, and is developing the safety and life-support equipment that would allow Dragon to double up as an astronaut taxi.
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