The two sides had until midnight to reach a deal and agreed on a contract reportedly worth around $10 million, the Nationals said on their website (washington.nationals.mlb.com) citing a baseball source.
Harper is a highly-touted 17-year-old with devastating power at the plate.
The Nationals plan to convert Harper from a catcher to an outfielder, though he is not expected to reach the Majors for two to three years.
Harper is the second successive No. 1 overall pick for Washington, who signed exciting pitcher Stephen Strasburg to a record contract just before last year's deadline.
Strasburg has overwhelmed opponents at times and has a 5-3 record with a 2.97 ERA.
(Reporting by Jahmal Corner in Los Angeles; Editing by Peter Rutherford)