The Spanish third seed was out-hit and out-fought for large periods of the match on Centre Court by his US opponent but found an extra gear when he needed to in a match lasting nearly four hours.
The two men were meeting for the first time since Alcaraz came out on top in an epic five-set match in the semi-finals of the 2022 US Open on the way to his first Grand Slam title.
"Always a big challenge playing against Frances," said Alcaraz. "He is a really talented player, really tough to face and he has shown again that he deserves to be at the top and fight for big things.
"Really difficult for me to adapt my game, to find solutions to put him in trouble but really happy to do it at the end of the match."
The Spaniard, bidding to become only the sixth man to win the French Open and Wimbledon back-to-back, beat Novak Djokovic in a thrilling final last year.
He will face America's Brandon Nakashima or 16th seed Ugo Humbert of France in the fourth round as he seeks a fourth Grand Slam title at the tender age of 21.
Tiafoe power
Tiafoe, seeded 29th, put Alcaraz's serve under intense pressure in the opening set, carving out six break points and taking two of them to win it 7-5.
But the Spaniard slashed his error count in the second set, breaking twice to level the match under the roof on a damp day in London.
Alcaraz fended off another clutch of break points from Tiafoe in the third set but cracked in the seventh game and the American, combining brutal power with finesse, made the crucial breakthrough, whipping up the crowd as he sealed the set.
The champion was in deep trouble at 0-30 in the ninth game of the fourth set but won four straight points to stay on serve.
An energised Alcaraz stepped up a level in the tie-break, powering his way to a 5-0 lead and cupping his ear to the crowd as he pulled level at two sets apiece.
An early break in the decider for Alcaraz emphatically underlined the change in momentum and he repeated the feat to take a 4-1 lead.
Tiafoe held serve to delay the inevitable but Alcaraz served out to love, sealing the deal with a clever drop shot before letting out a roar and raising his arms to the sky.
The three-time major winner said he had faced a lot of difficult moments in the pivotal fourth set but was just thinking about "hitting one more ball".
"I have to tell myself to go for it," he said. "If I lose it, I lose it but I have to feel that I want it all the time and most of the time it works on my side and that happened today again."
Alcaraz is chasing his third trophy of the season. Last month, he triumphed at Roland Garros to become the youngest man in history to win a major on all three surfaces.
