SpaceX founder Elon Musk said he was "overcome with emotion" as the company's rocket took astronauts Robert Behnken and Douglas Hurley into space.
The launch on Saturday, which sent the Nasa astronauts on a 19-hour flight to the International Space Station, was the second attempt after a first was called off on Wednesday due to bad weather.
Mr Musk said: “I think this is something that should really get people right in the heart – anyone who has a spirit of exploration.”
He added: “I am really quite overcome with emotion – it’s kind of hard to talk really.
SpaceX Falcon 9 launch - in pictures
“It’s been 18 years working towards this goal. It’s really hard to believe that it’s happened.
“This is a craft made by humans, for humans, I think it’s something humanity should be proud about occurring on this day."
The rocket - named the Falcon - took off at 8.22pm UK time on Saturday from Cape Canaverel in Florida.
There had previously been concerns that unfavourable weather could delay the launch again.
The mission marks the first time Nasa has launched astronauts from the US since 2011.
SpaceX also became the first private company to send humans into orbit.
Mr Musk said the mission is the “first step on a journey that would see humans become a multi-planetary species”.
But he warned against complacency, adding that the “return can be more dangerous than the ascent”.
He went on: “We need to bring them home safely and make sure that we are doing everything we can to minimise that risk of re-entry."
Donald Trump hailed the launch as the start of "a new age of American ambition."
The president added: “Under Nasa’s commercial crew programme, we will use rockets and spacecraft designed, built, owned and launched by private American companies at a fixed price for the American taxpayer.
“Today’s launch makes clear the commercial space industry is the future.”
The Falcon is due to dock at the International Space Station at 3.29 pm UK time, where Mr Behnken and Mr Hurley will join Nasa’s Chris Cassidy and Russia’s Anatoli Ivanishin and Ivan Vagner.
This mission is the last step towards Nasa clearing SpaceX's astronaut carrier the Crew Dragon for long-term manned missions to space.
