Four-time Grand Slam champion Alcaraz reached the fourth round at the Foro Italico for the first time after a battle with spirited Djere.
In the next round, Alcaraz will face Karen Khachanov who knocked out Italian wild card entrant Francesco Passaro 6-3, 6-0.
Alcaraz was playing in his second match since suffering a thigh injury in his Barcelona Open final defeat by Holger Rune last month.