The deal is worth $8.1 million, according to multiple reports, setting a record for draft bonuses. Pitcher Gerrit Cole, now with the Houston Astros, received an $8 million bonus from the Pittsburgh Pirates as the first pick in the 2011 draft.
The 21-year-old Rutschman, a switch-hitter, batted .411 with 17 home runs as a junior for the Beavers this year. He won the Golden Spikes Award, which recognizes the top amateur baseball player in the United States.
Baltimore had not had the first draft pick since 1989, when it selected pitcher Ben McDonald out of LSU.
The Orioles announced the signing on Monday. The deadline to sign draft picks is July 12.
The team plans to introduce Rutschman to the crowd at Camden Yards on Tuesday during a game against the San Diego Padres.
The Associated Press contributed to this report.