The Hurricanes (17-3, 8-0 ACC) had pulled to within one point after Durand Scott connected on 2-of-4 free-throw attempts over two consecutive possessions.
When Miami got the ball back with 10 seconds left, Shane Larkin drove to the hoop but missed a layup. Johnson, though, was there to tip in the rebound and hand the Hurricanes their ninth straight win.
Scott ended with 18 points, Julian Gamble added 16 and Johnson provided 15 with eight boards.
The Wolfpack (16-6, 5-4 ACC) got 18 points and 12 rebounds from C.J. Leslie, but it wasn't enough to avoid the loss.
Tyler Lewis added 16 points and five assists for N.C. State.