Richard Howell added 12 points and 12 rebounds for the Wolfpack (11-4), who wrapped up nonconference play with a five-game winning streak.
N.C. State limited the Hornets to 29.5 percent (18 of 61) shooting from the floor, including 23.5 percent in the first half.
N.C. State opened the game with a 9-2 run and never trailed.
The Wolfpack closed the first half with a 22-0 spurt, scoring on 11 of 12 possessions to take a 42-18 lead into halftime.
Delaware State missed nine consecutive shots while going scoreless for the final 7 minutes of the half.
Reserve guard Alex Johnson helped spark the surge for the Wolfpack. He made a driving layup and a 3-pointer on back-to-back possessions to give N.C. State a 29-18 lead with 5 minutes remaining.
Leslie then contributed a slam, worthy of a dunk contest, after getting a steal in the open court. He lobbed the ball to himself as he leaped, caught it in flight and finished with a two-handed jam that gave the Wolfpack a 34-18 lead.
The Hornets scored fewer than 60 points for the fifth consecutive game. They ended their scoring drought on their first possession of the second half when Walker made a put-back in transition.
Tate, Delaware State's leading scorer for the season, returned after serving a four-game suspension for conduct detrimental to the team. He came off the bench to shoot 5 of 20 from the floor.