The British number two led by a break in each set but Wimbledon champion Alcaraz fought back to win 7-6 (1) 6-4 after two hours and 27 minutes, booking his place in the fourth round.
Alcaraz went into the match having won all three of the previous meetings between the pair, although Evans claimed his first set against the Spaniard in another entertaining contest at the US Open last month.
A first set that lasted an hour and 27 minutes saw Evans withstand immense pressure in a contest of incredible rallies before finally succumbing to the top seed in a tie-break.
Evans led 4-1 but converted his only opportunity, while Alcaraz could take only one of his 13 break points, with his opponent saving six of them in a fifth game that lasted nearly 20 minutes.
Alcaraz fought back to make it 4-4 and dominated the tie-break but Evans did not allow his head to drop and came out strongly in the second set.
A break for 2-1 was immediately pegged back by Alcaraz, who found a second break of the Evans serve when he needed it most to claim victory on his second match point.
“It was one of the toughest matches I’ve played so far outside of the grand slams,” Alcaraz said on Amazon Prime Video. “Every time I face Daniel it’s a tricky match.
“I think both of us played a high quality of tennis. I’m really happy to get through.”
Additional reporting by PA Sport.