The two-man NASA crew blasted off on a Falcon 9 rocket at 8.22pm on Saturday – in the first ever space mission run by a private company.
Their Dragon capsule, operated by billionaire Elon Musk’s firm, attached to the ISS at 3.16pm UK time. It completed docking at 3.30pm.
More than two hours later, Mr Hurley and Mr Behnken floated safely into their new home.
This is the first time in human history @NASA_Astronauts have entered the @Space_Station from a commercially-made spacecraft. @AstroBehnken and @Astro_Doug have finally arrived to the orbiting laboratory in @SpaceX's Dragon Endeavour spacecraft. pic.twitter.com/3t9Ogtpik4— NASA (@NASA)May 31, 2020
They were greeted by fellow American Chris Cassidy as well as two other space station residents, Russia cosmonauts Anatoli Ivanishin and Ivan Vagner.
The new arrivals are now officially members of the Expedition 63 crew.
Speaking aboard the space station, Mr Hurley said: “It obviously has been an honour just to be a small part of this. We have to give credit to SpaceX, the Commercial Crew Programme and, of course, Nasa.
“It’s great to get the United States back in the crewed launch business and we are just really glad to be on board this magnificent complex.”
Meanwhile back on Earth, Nasa administrator Jim Bridenstine said: “The whole world saw this mission and we are so, so proud of everything you’ve done for our country and, in fact, to inspire the world.”
Mr Behnken said during their time on the ISS, the pair hope to put the Dragon capsule – which they have christened Endeavour – through its paces as well as assist other members of the crew in various other tasks.
He said: “Endeavour is going to get a lot of checkout over the next week or two here and hopefully we will be able to declare her operational.
“Doug and I will be able to take some burden of Chris and his crew mates Ivan and Anatoli so that we can keep the space station operating at a peak possibility.
"So we are looking forward to contributing any way that we can and like I said trying to keep (the) space station as productive as possible.”
“We’re just happy to be here and Chris [Cassidy] is going to put us work. And hopefully we will fit in and not mess too many things up.” @Astro_Doug on him and @AstroBehnken being the newest crew members of the @Space_Station. pic.twitter.com/Y5xZJFn2As— NASA (@NASA)May 31, 2020
The spacecraft had to perform a series of manoeuvres to get close enough to dock at the space station on Sunday afternoon.
Mr Hurley and Mr Behnken took over the controls and briefly piloted the Dragon before putting it back into automatic for the final approach.
Mr Hurley said the capsule handled "really well, very crisp."
Shortly after docking, but before they entered the hatch to the ISS, Mr Cassidy greeted his colleagues by ringing the ship's bell.
He then announced: “Dragon arriving.
“The crew of Expedition 63 is honoured to welcome the Dragon and the Commercial Crew Programme.
“Bob and Doug, glad to have you as part of the crew.”
The mission, named Demo-2, marks the first time NASA has launched astronauts from US soil in nine years.
Its aim is to show SpaceX’s ability to ferry astronauts to the space station and back safely.
It is the final major step required by the Crew Dragon to acquire NASA certification for more long-term manned missions to space.
British astronaut Tim Peake celebrated Sunday's landmark moment on Twitter, writing: “Another milestone in the history of human spaceflight.
"Congratulations @SpaceX and welcome to the @Space_Station @AstroBehnken and @Astro_Doug.”
— Tim Peake (@astro_timpeake) May 31, 2020
A few hours before docking, Mr Hurley and Mr Behnken reported that the capsule was performing beautifully.
Just in case, they got back into their pressurised launch suits and helmets ahead of their arrival.
NASA has yet to decide how long the pair will spend at the space station – between one and four months.
While they're there, the veteran astronauts will join their fellow ISS residents in performing experiments and possibly spacewalks to instal fresh station batteries.
Welcome aboard the @SpaceX Crew Dragon spacecraft!
In this video from space, @AstroBehnken and @Astro_Doug reveal the name of their capsule: Endeavour. Take a look inside as the crew continues their journey to the @Space_Station: https://t.co/K9S5mejONx pic.twitter.com/mvH8UhE5FW— NASA (@NASA)May 31, 2020
In a show-and-tell earlier on Sunday, the astronauts gave a quick tour of the Dragon's interior.
The blue sequined dinosaur accompanying them – their young sons' toy, named Tremor – was also in good shape, Mr Behnken assured viewers.
Tremor will now join Earthy, a stuffed-toy globe delivered to the space station on an unmanned Dragon test flight last year.
Mr Behnken said both toys would return to Earth with them upon the mission's completion.
After lift-off on Saturday, Mr Musk told reporters that the capsule's return will be more dangerous in some ways than its launch.
In an emotional message, he said: "This is hopefully the first step on a journey toward a civilisation on Mars."