The top-seeded Spaniard breezed through his opening matches at Flushing Meadows and appeared on track for another easy win in Saturday's third round tie, but had to find his best form after Evans mounted a third-set comeback.
Alcaraz next plays Italian Matteo Arnaldi, who also beat a Briton. The Italian world No. 61 defeated 16th seed Cameron Norrie 6-3 6-4 6-3.
"He's a tricky opponent," Alcaraz said of Evans. "We made great points, great shots. A lot of different kinds of situations."
Alcaraz moved through the first set like a freight train, winning the first four games, and the 26th-seeded Evans failed to convert any of three break points in the eighth game.
Evans did let out a roar as he broke Alcaraz with an unreturnable backhand down the line in the second game of the second set but the Spaniard broke back immediately and the Briton lost another break in the fifth.
Alcaraz broke Evans again with a well-executed drop shot to close out the second set. However, Evans upped his level in the third set, channelling his frustration into a superb game seven where he broke with a backhand winner.
The 20-year-old Alcaraz tapped into his superior speed and agility to tame Evans, 13 years his senior, in the fourth set, zipping back and forth along the baseline before breaking with a sublime forehand winner in the sixth game.
He finished the entertaining clash with a forehand winner, one of 27 in the match, prompting cheers of approval from the rapt crowd inside Arthur Ashe Stadium.
Jack Draper, 21, offered British fans hope as he outlasted American wildcard Michael Mmoh 6-4 6-2 3-6 6-3 to book his first trip to a grand slam fourth round.
He will face the winner of the tie between Russian eighth seed Andrey Rublev and France's Arthur Rinderknech.
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