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.
And the Spaniard was not at his best against Djere whose three career tournament wins have all come on clay, his most recent title in March, at the Chile Open in Santiago.
The 22-year-old was broken twice in the first set which looked like going either way until he took control in the eventual tie-break, before sweeping Djere aside in the second.
"I just thought that I had to stay there mentally strong, waiting for my chances because I was playing great," said Alcaraz.
"I was playing great, moving, hitting the ball. It was just a few mistakes that I made."