"Soft capture" - the first part of the docking procedure - was completed at 11:01 pm (0401 GMT Tuesday), with the second part of the procedure set to happen a few minutes later.
The spacecraft spent 27 hours getting there after launching Sunday night (EST).
"For the first time in history, there is a commercial capability from a private sector entity to safely and reliably transport people to space", said Phil McAlister, director of commercial spaceflight development at Nasa.
Baby Yoda tags along on SpaceX's first operational mission for NASA
Hopkins and his crew - Victor Glover, Shannon Walker and Japan's Soichi Noguchi - will join two Russians and one American who flew to the space station last month from Kazakhstan.
Quite similar to Crew Dragon's flawless Demo-2 astronaut launch debut, the Crew-1 spacecraft is scheduled to arrive at what is known as the ISS keep-out sphere around half an hour after its final major course-change thruster burn.
The three Americans and one Japanese astronaut will remain at the orbiting lab until their replacements arrive on another Dragon capsule in April.More news: Johnny Sexton and Robbie Henshaw ruled out of England clash
Mr Glover's inclusion in this mission, called Crew-1, has its own historic significance. Glover will be the first African-American to move in for a long haul.
"We hope you enjoyed your first night aboard Dragon", SpaceX flight controller Jay Aranha told Crew Dragon commander Mike Hopkins, speaking on a line from the company's Hawthorne, California, headquarters.
Though mission control had to work through some minortechnical hang-ups on the autonomous Crew Dragon spacecraft overnight, the ride appeared to be going smoothly yesterday.More news: Covid-19 may have appeared in Italy earlier than thought
On their way to the station, the astronauts broadcast a tour showing off the high-tech Resilience, including a plush Baby Yoda bouncing around the capsule as a zero-gravity indicator.
Meanwhile, the astronauts were able to swap their spacesuits for more comfortable clothing and get some sleep.
Before returning home, they'll be joined by yet another group of astronauts on a mission dubbed Crew-2 that's due to launch in the USA spring.More news: HBO Max Ends Amazon Standoff, Reaches Deal for Fire TV Devices