But it's a very different type of vehicle to Nasa's retired spaceplane.
With the on-time liftoff by the 260-foot rocket, SpaceX, founded by Musk, the Tesla electric-car visionary, became the first private company to launch people into orbit, a feat achieved previously by only three governments: the U.S., Russian Federation and China.
But the company had to walk before it could run. The spacecraft is able to dock with a newer automated global docking adapter installed on the ISS, unlike the original cargo version of Dragon, which required manual capture by the robotic Canadarm 2 controlled by astronauts on the station.
A SpaceX Falcon 9 rocket carrying the Crew Dragon spacecraft lifts off in Florida.
From launch up until shortly before re-entry, the capsule is attached to a section called the trunk which has solar panels, heat-removal radiators and fins to provide stability during emergency aborts.
"Dragon's a slick vehicle", Behnken said. Each Draco is capable of producing 90 pounds of force in the vacuum of space.
"Probably one of the biggest features of Dragon are the touchscreens on the inside. That was the beginning of what we all got to experience today". Both companies are focused on test missions, including abort system demonstrations and crew flight tests, ahead of regularly flying crew missions to the space station. Hurley admits that the "glass cockpit" took a bit of getting used to.
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"Without further ado, we would like to welcome you aboard capsule Endeavour", Hurley said.
"I really think it's a demonstration of the trust they had both in the NASA team and the SpaceX team to get them safely to orbit", he said.
US-Russia cooperation is not expected to end once Crew Dragon goes into service. The updated cargo Dragon and Crew Dragon are created to work with the new automated system.
Bridenstine cautioned, however, saying "we're not celebrating yet".
During their stay, Bob Behnken and Doug Hurley will perform more checks on Endeavor to certify its readiness as the United States transitions to using the commercial sector for rides to the ISS.
"What should have happened a long time ago happened", he added.
SpaceX conducted a successful test flight of Crew Dragon to the ISS in March 2019 with a sensor-laden mannequin on board named Ripley, after the character played by Sigourney Weaver in the "Alien" movies.
"We were filled with the sense of pride and unity that brings us together as Americans", Trump said.
On the final flight of the space shuttle, the crew chose an eclectic mix, including Elton John's "Rocket Man", R.E.M.'s "Man on the Moon" and "Here Comes the Sun" by the Beatles.
There's a minor chance that the spacecraft's asymmetric design - driven by the placement of its emergency escape system - could cause it to roll too much.