Bautista Agut came back from a set down to beat Hungarian qualifier Marton Fucsovics 4-6 7-6(1) 6-1. Fucsovics was close to claiming the biggest win of his career in the second set before Bautista Agut forced a tiebreak.
The third set was plain sailing for the Spaniard and a double fault at 5-1 gave Bautista Agut a second match point, which he converted with a sliced backhand after an engrossing rally at the net.
Fucsovics hit 40 winners but the 41 unforced errors proved to be his downfall.
"Next round is going to be a Spanish round against Pablo," Bautista Agut said after the win. "Hopefully one of us can take the title."
In the other encounter, Carreno Busta needed two hours and 45 minutes to dispatch Frenchman Adrian Mannarino, winning 7-5 6-7(3) 6-4.
Mannarino fired 12 aces but he was wayward with his second serve while Carreno Busta, who had 17 chances to break serve, only managed to do so on four occasions.
On match point in the third, a 20-shot rally was finished by Carreno Busta when he fired a forehand winner to advance and set up a last-16 clash with his compatriot, who has beaten him twice before in three encounters.
Bulgaria's Grigor Dimitrov edged out Canadian eighth seed Denis Shapovalov 6-3 7-6(3) in a scrappy first round encounter at the Rotterdam Open on Monday.
The world number 22 admitted he never hit top gear but that did not stop him from setting up a last 16 meeting with either Canadian Felix Auger-Aliassime and Germany's Jan-Lennard Struff.
"There was not a lot of rhythm," Dimitrov, who was beaten by Roger Federer in the 2018 final, told reporters. "There were not too many rallies, but I was able to stay in control.
"I was calm in the important moments and came to the net when I had to."
British number one Dan Evans claimed a 6-3, 7-5 victory over Germany's Philipp Kohlschreiber, breaking serve in both sets and serving eight aces against the former Rotterdam semi-finalist.
He will now face either Karen Khachanov or Fabio Fognini, seeded fifth.
Djokovic edges Thiem in five-set thriller