He took just over two hours to beat Bautista Agut who was bidding to reach his second Wimbledon quarter-final in his 10th appearance here.
Paul, whose grass game is finely tuned after he won the Queen's Club warm-up event last month, had just too much power and variety for the veteran, now ranked 112.
Bautista Agut made the American work in the second set but a couple of unforced errors lost him the chance to draw level and Paul raced through the final set, clinching the match with an ace.
The players had arrived on court late because of rain showers and the shadows were long across the court when they finished.
"It worked out great," Paul said. "I always play good tennis on this court.
"I know the weather hasn't been cooperating but maybe my next match will be inside," he joked.
A match-up with Alcaraz is likely to be scheduled on Centre Court which has a roof.
"He (Alcaraz) plays amazing tennis on grass but I'm playing pretty good too," Paul, who is on a nine-match winning streak, added.
