Ewing scored 18 of his 19 points in the first half to help Duke build a 10-point lead in the first eight minutes.
Shavlik Randolph led the way with 20 points, helping Duke advance to a second-round encounter with No. 8 Seton Hall.
Malcolm Campbell scored 26 points for Alabama State, the lightly regarded Southwestern Athletic Conference champion. The Hornets fell to 0-2 all-time in the NCAA Tournament.
Five players scored in double figures for the Blue Devils, who are seeking their third Final Four appearance in the last six years. Alabama State was making just its second NCAA Tournament appearance.
"I think we were trying to conquer the world in the first half," Duke Coach Mike Krzyzewski said. "You need to have fun, too. They have been under a lot of pressure all year and finally in the second half they settled down and played basketball."