The 6-foot-9, 205-pound Flagg - the nation's top-ranked recruit and a preseason Associated Press All-American - made 6 of 15 shots and all six of his free throws.
Fellow rookie Kon Knueppel powered the offense with 22 points and three 3-pointers for the Blue Devils (1-0), who shot 55.6% for the game.
Quion Burns scored 15 points for the Black Bears (0-1), which shot just 6 for 24 (25%) after halftime as Duke pulled away.
The Black Bears were picked fourth in the America East, their highest projected finish since the 2010-11 season. They hung around in the first half (45.5% shooting) but everything was a struggle after the break.
Duke, the Atlantic Coast Conference favorite, had six players score in double figures, including returnees Caleb Foster (11) and Tyrese Proctor (10).
Duke pulled away with an 11-0 run in the second half. Knueppel made a corner 3 off a feed from Flagg and Sion James scored twice in the paint to push the margin to 60-39 with 14:33 left.
Knueppel, a 6-7 wing, had 15 points on 5-for-8 shooting with a pair of 3s by halftime. That included a 4-point play in transition that had him outscoring Maine 13-9 in those opening 7 1/2 minutes as Duke ran out to a double-digit lead.
The Blue Devils host Army on Friday in a home tuneup before facing No. 23 Kentucky in next week's Champions Classic in Atlanta. The Black Bears return home Wednesday to host Maine-Fort Kent, a United States Collegiate Athletic Association member.
The Associated Press contributed.