Since the shuttle's retirement, the US has been forced to rely on Russia to take its astronauts to the station, an expensive and troubling arrangement that now costs US$76 million ($112 million) a seat.
By relying on contractors, the US will save nearly US$20 million a flight, Bolden said, and allow more astronauts to fly at a time.
The first flight, expected near the end of 2017, had been pushed back from this year because of funding delays. But now even the revised timeline is in jeopardy, some fear, after Congress slashed US$300 million from the programme.
If the cuts stand, they will delay the programme by two more years, Senator Bill Nelson, a Democrat for Florida, has said.
Bolden has also blasted Congress, saying: "By gutting this programme and turning our backs on US industry, Nasa will be forced to continue to rely on Russia to get its astronauts to space - and continue to invest hundreds of millions of dollars into the Russian economy rather than our own."
There are also renewed concerns about whether it is prudent to rely so heavily on contractors for such important missions.
Both Orbital Sciences (now Orbital ATK) and SpaceX, the companies used by Nasa to fly food and supplies to the station on unmanned missions, have seen their rockets explode in recent months.
And now, both companies are temporarily sidelined as they work to fix whatever problems caused the failures.
Despite the explosions, though, Nasa has said it remains confident in the companies and is moving ahead with its plans. Bolden called the programme "a worthy successor to the incredible three-year run of the space shuttle programme".
The astronauts chosen by Nasa to fly on the upcoming missions are among the agency's best and most experienced.
The astronauts
The four selected for the next United States mission are:
Sunita Williams
A navy captain and a Naval Academy graduate. A helicopter pilot, Williams deployed during the Persian Gulf War and later became a helicopter test pilot instructor. Selected to be an astronaut by Nasa in 1998, she has spent a total of 322 days in space over two missions and has spent more time on spacewalks - 50 hours and 40 minutes - than any other female astronaut.
Douglas Hurley
A retired Marine Corps colonel who became the first Marine test pilot to fly the F/A-18E/F Super Hornet. He has flown two space shuttle missions to the space station, one on Endeavour, the other on Atlantis, and has spent a total of 683 hours in space.
Eric Boe
A colonel in the air force who flew 55 missions monitoring the airspace over southern Iraq after the Persian Gulf War. On his first flight to the space station in 2008, he helped expand the station's living quarters and delivered a new bathroom, kitchen and exercise machine. During his second mission aboard space shuttle Discovery, he orbited Earth 202 times.
Robert Behnken
Also an air force colonel. Behnken served as an F-22 test pilot and flew two space shuttle missions, the first in 2008, the second two years later. He's performed a total of six spacewalks.