The Trail Blazers had forged a 108-108 tie with 10.9 seconds remaining on Damian Lillard's 3-pointer, but the Pelicans quickly responded as head Coach Monty Williams declined to call a time-out.
Evans got the ball on the right side and shook off Mo Williams' defense to nail the decisive score.
Portland had one last chance but LaMarcus Aldridge missed a jump shot at the buzzer.
Jrue Holliday had 31 points, 13 assists and seven rebounds for New Orleans, which improved to 8-2 in games decided by four points or less.
Anthony Davis chipped in 27 points and 12 rebounds and Evans finished with 20 points.
Lillard netted 29 points, including five treys, and Aldridge added 28 with eight rebounds for the Blazers, who dropped a second straight.