In a game of seven yellow cards, Napoli went ahead in the 15th minute when Lorenzo Insigne's cross was parried by Sassuolo goalkeeper Gianluca Pegolo and the ball hit Milik on the thigh before rebounding into the net.
Manuel Locatelli had the ball in the net for the visitors 10 minutes before halftime but it was disallowed for handball in the build-up following a VAR review.
Sassuolo threatened again early in the second half, Domenico Berardi curling a shot over the crossbar and Kalidou Koulibaly's tackle preventing Alfred Duncan from equalising.
Napoli, who face AC Milan in the last eight, wrapped up the game when Milik got away on the left and pulled the ball back for Fabian Ruiz to score the second in the 74th minute.
(Reporting by Brian Homewood, editing by Ed Osmond)
