Daily services will start on 23 May.
They will supplement United's existing flights between Edinburgh and Newark in New York, launched in 2004, which now compete against Delta and American Airlines.
The airline has also flown between the Scottish capital and Chicago since 2014.
However, United will reduce its year-round Glasgow-Newark flights to summer only from next month for the first time since it launched the route 19 years ago, as The Scotsman revealed in July.
News of the Washington DC route comes a day after Edinburgh announced its busiest ever August and second busiest month ever, when it handled 1.4 million passengers.
The further growth of Scotland's busiest airport comes despite its repeated calls for air passenger duty to be reduced, with ministers planning to halve the tax next year.
Edinburgh Airport chief executive Gordon Dewar said: “This is fantastic news for Edinburgh Airport and Scotland as we connect our two capital cities for the first time.
“As Scotland’s busiest airport, we are a leader for international destinations, and this demonstrates the confidence of US carriers in the Edinburgh market, standing ahead of other parts of the UK with such strong inbound tourism demand.
“Our partners at United have seen first-hand the positive impact of flights from Edinburgh to New York Newark and Chicago, and we are keen to replicate that to Washington for both a business and tourist audience."
Transport minister Humza Yousaf said: "The launch of a direct route into America’s capital city is great news for Scottish businesses, giving them better access to important US markets through United’s hub operation.
"It will also help support 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."