The 20-tonne vehicle hooked onto the back of the 415km-high orbiting platform at 14:07 GMT.
It is carrying food, water, equipment and fuel for the six live-aboard astronauts on the ISS.
Albert Einstein will also provide some useful extra space during its four-month stay, and use its engines to push the platform higher into the sky.
The station has a tendency to drift back towards the Earth over time, and the European freighter with its powerful thrusters can provide an altitude boost.
They had the ability to command the vehicle to retreat if they had any concerns about its behaviour, but Albert Einstein performed flawlessly, docking over the Pacific Ocean just east of Japan.
It is the fourth such vehicle produced by Esa and European industry. The cargo ships form part of the subscription Europe pays to belong to the ISS project.
One more vehicle is planned to fly next year before production ceases at Bremen, Germany.
The ATV assembly line will then be turned over to building a propulsion unit for US space agency’s (Nasa) future crew ship, Orion.
This capsule will carry astronauts beyond the space station to destinations such as asteroids and Mars. It will need a “service module” to push it through space and Nasa has engaged Esa to adapt ATV technology for the purpose.
Albert Einstein will stay attached to the ISS until late October.
Astronauts will gradually remove its “Russian water”, air and 2.5 tonnes of dry cargo, starting on Tuesday next week. It will all be replaced with rubbish that has built up on the platform.
When the freighter leaves the station, it will take this refuse on a destructive dive into the Earth’s atmosphere.