The Spanish world number three, seeking a fourth Grand Slam title, beat his 12th-seeded opponent 5-7, 6-4, 6-2, 6-2 under the No. 1 Court roof.
Alcaraz was slow out of the blocks, broken twice by the in-form American in a 72-minute first set.
Paul raced into a two-game lead in the second set as Alcaraz again faltered on serve but the Spaniard got back on level terms and broke again to level the match.
Three breaks in the third set put the world number three in charge and he stepped up another gear in the fourth set, winning 92 percent of first-serve points and making just four unforced errors.
Alcaraz, seeking to become just the sixth man to capture the French Open and Wimbledon titles back to back, said it had been a tough start to his quarter-final against the recent champion at Queen's.
"If I'm struggling a little bit to find the solutions, if the opponent is playing great tennis and I don't get him in trouble, I believe at the end that I'll be able to come back and be able to find solutions," he said.
Looking forward to his match against Russian fifth-seed Medvedev, whom he beat last year on his way to winning his first Wimbledon title, he said: "Hopefully I'm going to get the same result.
"He just won against Jannik Sinner, the best player right now, so I know that he's in really good shape so I have to play my best, I have to believe in myself and try to keep going if I want to beat him."
