Leslie's partner in the Wolfpack (5-2) front court, Richard Howell added 13 points and 10 rebounds despite missing part of the first half due to a blow to the head from an elbow.
Lorenzo Brown also scored 16 points for North Carolina State, who had lost two of their last three coming into Tuesday's game.
Shabazz Napier scored 19 to lead UConn (6-2), who had their two-game winning streak broken. Ryan Boatright had 18 points off the bench and Enosch Wolf picked up 10 points and seven rebounds.
North Carolina State led 32-31 at the half and did not put the Huskies away until Scott Wright hit the second of two free throws with 31 seconds to play. The Wolfpack was 13-for-21 from the free-throw line for the game.