The carrier said the year-round flights, from £138 return, would serve smaller airports outside Boston and New York City.
They will start with daily services to Stewart airport, some 70 miles north of New York City, from 15 June.
A day later, four flights a week will begin to Providence, Rhode Island, 60 miles south of Boston.
Three weekly services will start from 17 June to Hartford, Connecticut - midway between Boston and New York.
Norwegian chief executive Bjorn Kjos announced last May that the airline would launch transatlantic routes from Edinburgh, including to Boston, as The Scotsman and Edinburgh Evening News were the first to report.
He indicated then that west coast flights, such as to San Francisco, would follow once Norwegian took delivery of longer-range Boeing 787 Dreamliner aircraft.
The fares - from £69 one way - are likely to far less than those currently charged by American, Delta and United on the Edinburgh-New York route.
Edinburgh Airport chief commercial officer John Watson told The Scotsman last year that Delta joining the competition last summer meant typical return fares of £800 being halved.
Norwegian is expected to prove cheaper, even though passengers will have to pay for hold luggage and food and drink, along with potentially higher transport charges from the remoter airports.
Mr Kjos said today: “Norwegian is shaking up transatlantic travel once again with ground-breaking fares and never before seen routes.
"The cost of transatlantic travel has been too high for too long, so by connecting Scotland with smaller US airports, we can offer some truly affordable fares, allowing as many people as possible to fly.
"These new transatlantic routes from Edinburgh builds on the success of our long-haul flights from London into major US cities as we’re now able to offer passengers an even greater choice of affordable travel to the USA.”
Edinburgh Airport chief executive Gordon Dewar said: “These new low-cost direct services between Scotland’s capital city and the east coast of America are a real game-changer – making travel between these two parts of the world easier than ever before and opening up possibilities to more business and leisure passengers than ever before.
Transport minister Humza Yousaf said: “As Edinburgh’s first low-cost, long-haul carrier, these new flights will help build on Scotland’s strong business links with the USA.
"They will also help boost our tourism industry by making it even easier for visitors to get a taste of the fantastic experiences Edinburgh and Scotland as a whole have to offer.
"I have no doubt this is the first step in a successful new relationship between Edinburgh Airport and Norwegian, and we look forward to helping them build on that partnership in the future.”
The flights will be operated by brand-new, single-aisle Boeing 737 MAX aircraft.
The airline announced in January the new routes would create more than 130 jobs in Edinburgh - at least 100 cabin crew and 30 pilots.
It already flies six European routes from the capital.