Maurice Kemp, who had 10 points, put the 10th-seeded Pirates (15-15) ahead 66-64 on a layup with 57 seconds left -- the game's 13th lead change.
Paul missed a free throw for East Carolina, but Lucas Kuipers couldn't hit a tying jumper for Rice with 4 seconds remaining.
Paris Campbell added 12 points for East Carolina, making 4 of 9 3-pointers. The Pirates will play second-seeded Southern Mississippi in the quarterfinals Thursday.
Kuipers and Kazemi scored 10 points each for Rice. Kazemi had 12 rebounds, giving the junior his 44th career double-double.