Freshman Brandon Ingram added 25 points for the Blue Devils (15-5, 4-3 ACC), who trailed by 10 points late in the first half but seized momentum with their go-ahead 14-0 run.
Matt Jones came up with two 3s in the game-turning burst, which allowed Duke to build a double-digit lead and put the Wolfpack (11-9, 1-6) into catch-up mode.
Duke kept the pressure on, too, shooting 71 percent from the field after halftime — including 7 for 11 from 3-point range — in an efficient showing and finished at 54 percent for the game.
Anthony "Cat" Barber scored 19 points to lead N.C. State, which shot 52 percent but couldn't keep up once Duke's offense found its flow.
It had been a rough two weeks for the reigning NCAA champions, whose reshaped roster had been short-handed ever since senior Amile Jefferson went out with a foot injury. First came a loss at Clemson, followed by last weekend's loss in a shootout against Notre Dame.
Then came Monday's home loss to Syracuse, the program's second straight at its famously hostile Cameron Indoor Stadium. It was the program's first three-game skid since 2006-07 and might ultimately knock Duke out of the AP Top 25 for the first time since that season.
Winning at N.C. State — where the Blue Devils had lost three of four — eased some of that frustration.
Allen carried the offense early with 16 first-half points, while Ingram finished 10 for 16 from the field to go with seven rebounds. And Jones' two 3-pointers during the 14-0 burst were his only baskets of the game, helping Duke turn a 56-52 deficit into a 66-56 lead with 9:24 left.
The Wolfpack got no closer than seven points again.
The game was played a day after the area received a wintry mix of snow, sleet and freezing rain to create treacherous driving conditions in some nearby areas. North Carolina Gov. Pat McCrory discouraged fans earlier Saturday from trying to attend and told them to "enjoy the game on TV," though that didn't put much of a damper on a home crowd that was its normal rowdy self.
Duke: Freshman Luke Kennard scored 12 points. ... Duke finished with a 31-28 rebounding advantage after being beaten on the boards in each of its past three losses. ... Duke made 14 of 18 free throws.
N.C. State: Abdul-Malik Abu and Caleb Martin each scored 14 points. ... Cody Martin had 13. ... N.C. State shot 57 percent in the first half.
Duke travels to No. 15 Miami on Monday.
N.C. State hosts Georgia Tech on Wednesday