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.
The Spaniard saw three set points come and go before having to save two match points, the second of which came courtesy of Sinner taking a wild swipe off a second serve.
He converted a fourth set point, much to the delight of the Centre Court crowd.
It was tough on Sinner, who faced just one break point over the first three sets.
He then saved two more in the opening game of the fourth set before carving out a crucial break himself for 3-1 when Alcaraz double-faulted.
Sinner saved another three break points in the fifth game before failing to cash in on three more match points in the eighth game.
Alcaraz had one more chance to break — his sixth of the fourth set — but he could not convert and that was his last opportunity.