South Africa's Louis Oosthuizen finished joint second.
Spieth, 21, is the youngest player to win the US Open since Bobby Jones in 1923.
He is the fourth-youngest player to win two majors and only the sixth to win the Masters and US Open in the same year.
"I'm in shock but I feel for Dustin," said Spieth. "It's cool to be able to have two legs of the grand slam."
Johnson, who started the final round in a four-way tie for the lead with Branden Grace, Jason Day and Spieth, said: "I did everything I was supposed to do. I hit the ball really well. I just really struggled getting it in the hole. I didn't think I was hitting bad putts, they just weren't going in."
For all of Johnson's heartbreak, the attention will be on the Spieth's chance to win all four majors in a calendar year.
The greats of the game - Ben Hogan, Arnold Palmer, Jack Nicklaus, Gary Player and Tiger Woods failed to achieve that feat.
The next test will come in July at the British Open at St Andrews. If Spieth succeeds in Scotland, then the expectation will swell during his four rounds at Whistling Straits in Wisconsin for the PGA Championship in August.
"I'm just focused on British Open now," said Spieth after his victory at the US Open. " I think that the Grand Slam is something that I never could really fathom somebody doing, considering I watched Tiger win when he was winning whatever percentage of the majors he played in and he won the Tiger Slam but he never won the four in one year.
"And I figured if anybody was going to do it, it would be him, which he still can.
"I think we'll just use that secret formula and see if we can maybe have it on the two weeks that the next two majors are held."
Spieth's consecutive wins mean that the four golfing majors are held by two men - himself and 26-year-old Rory McIlroy who won the British Open and PGA Championship last year.
With the former emperor Woods clearly declining - he missed the cut at Chambers Bay - the limelight is on Spieth and McIlroy - both fresh-faced characters who can enthuse a new generation.
And with the American 26-year-old Rickie Fowler holding the Players Championship - considered on the tour as the fifth major - Spieth says the game has a bright future.
"It's kind of cool to have two players holding the four majors and Rickie having the Players Championship. It's awesome that the game is in young hands.
"I don't think there is much of a rivalry with Rory. I've said that from the beginning. Rory has four majors and dozens of wins and I'm just starting out.
"I'm just happy to have the Masters and the US Open and to be chasing that number one spot that Rory holds. So I'm certainly focused on that."
Organisers of the US Open faced another torrent of abuse on Sunday after the final rounds.
On Friday Gary Player, whose nine majors came at all four grand slam tournament venues, lambasted the state of the course.
"This has been the most unpleasant golf tournament I’ve seen in my life," the 79-year-old South African told the Golf Channel. "I mean, the man who designed this golf course had to have had one leg shorter than the other. It’s hard to believe you see a man miss the green by one yard and the ball ends up 50 yards down in the rough."
On Sunday evening Ian Poulter from England was among numerous players who added their complaints. Poulter published a picture on social media showing jagged blades of grass on one of the greens.
"This was the surface we had to putt on," he wrote. "It is disgraceful that the USGA hasn't apologised about the greens. They were simply the worst, most disgraceful surfaces I have ever seen on any tour in all the years I have played."
"The US Open is a great tournament with incredible history," Chris Kirk, after finishing last, lamented. "The USGA should be ashamed of what they did to it this week. My score has nothing to do with why I feel that way, I played poorly. The course wasn't overly difficult, just tricked up."
"It's the kind of course I'd like to come and play with my mates, with a cart and some beers," Lee Westwood added.