The Dreamliner had originally been scheduled for delivery in 2008, but Boeing has suffered a string of setbacks, the latest being an onboard fire during test flights in January.
The fuel-efficient plane is made from lightweight composite materials.
Boeing plans to make 10 of the planes a month from 2013.
The plane was presented to ANA at a ceremony in Everett, Washington. On Tuesday it will fly to Tokyo.
Boeing chairman and chief executive James McNerney said: "Today we celebrate a significant moment in the history of flight."
In a light-hearted nod to the delays that have befallen the 787 project, he thanked ANA's president Shinichiro Ito for "waiting for this day".
Mr Ito replied that he knew "the road that led to today was full of great difficulties. Yet all of those challenges were overcome."
'Long journey'
Boeing says the twin-aisle, mid-size plane features the industry's largest windows, with higher cabin humidity and cleaner air - all of which combine to allow passengers to arrive at their destinations more refreshed.
But the problems with the Dreamliner have damaged Boeing's reputation, and the company will hope a successful launch will help put to bed some of the memories of the delays it suffered.
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Boeing delivers first Dreamliner
Boeing's vice president of marketing, Randy Tinseth, told the BBC: "This is a programme that has been a long journey. We believe it is going to be a great aeroplane.
"Ultimately we see the potential of thousands of 787 orders in the future."
Questioned on the delay to the 787 programme, Mr Tinseth added that "there are risks with every new aeroplane".
He said: "We've spent more money on this aeroplane than we anticipated, but again we're still in a position where we are not in a forward loss situation, we believe the programme will continue to be profitable."
Production of the Dreamliner is currently running at about 2.5 planes a month.
Airbus rival
So far Boeing has 821 orders for the 787, which it says is 20% more fuel efficient than similar-sized current planes. Seating a maximum of 290 passengers in the largest 787-9 version, the 787 is much smaller than Boeing's 747 jumbo jet.
However, Boeing considers that the 787 will prove popular with airlines, as it will enable them to fly directly to more smaller airports. Boeing's European rival Airbus is currently developing a direct competitor to the 787, the Airbus A350 XWB.
Airbus has more than 550 orders for the A350 XWB, but the plane is not due to enter service until 2013.
ANA intends to start flying its first 787 on a domestic route between Tokyo and Okayama-Hiroshima on 11 November. It will then put the plane on an international service from Tokyo to Frankfurt in Germany in January.
"A lot of carriers are betting that this Boeing 787 is going to be a winner," says George Hamlin, president of Hamlin Transportation Consulting.