Duke (13-1) won its fifth straight since suffering its lone loss at Michigan.
Davidson's Stephen Curry, the nation's leading scorer, recorded 29 points along with eight rebounds. But he hit just one three-pointer in eight attempts and turned the ball over seven times.
The Wildcats (10-3) also got 15 points from Andrew Lovedale. All of Davidson's losses this year have come against teams ranked in the top 25.
Duke built a 45-24 lead early in the second half, but Davidson went on an 11-0 run and eventually closed within 69-61 with 3:53 to play.
A key offensive rebound and three-pointer by Scheyer halted Davidson's comeback.
The Blue Devils have beaten Davidson 22 straight times.