The 20-year-old triumphed 6-1, 6-4, 6-7 (8/10), 6-3 to become the youngest man in the last-eight since Nick Kyrgios in 2014.
Sinner had never won a grass court match until this year's Wimbledon. But now he goes on to face either defending champion Novak Djokovic or Tim van Rijthoven for a spot in the semi-finals.
The 10th-seeded Italian needed six match points to seal the deal while 19-year-old Alcaraz was left to regret failing to convert six break points in the fourth set. "Carlos is a very tough opponent and a nice person.
It's always a huge pleasure to play him," said Sinner after making the quarter-finals of a Grand Slam for the third time. Sinner swept through the first two sets and appeared on course for a routine win until Alcaraz played a thrilling third set tie-break.