Mahut was the titleholder last year when Bautista Agut beat him in the quarterfinals en route to his maiden title here.
On Thursday, Mahut broke him five times to move into the quarters.
Second-seeded David Goffin of Belgium beat Jurgen Melzer of Austria 7-6 (2), 6-2 for his first win here in three attempts, and No. 6 Adrian Mannarino defeated Swiss qualifier Marco Chiudinelli 6-3, 6-2.
No. 8 Joao Sousa of Portugal, a semifinalist last year, lost to qualifier Illya Marchenko of Ukraine 6-3, 7-6 (2).
In women's play, second-seeded Jelena Jankovic reached the last eight with a 6-1, 6-4 win against Andrea Hlavackova of the Czech Republic, and next faces Annika Beck of Germany, who beat another Czech, Klara Koukalova, 6-4, 6-4.
Fourth-seeded Belinda Bencic beat French Open quarterfinalist Alison Van Uytvanck of Belgium 6-3, 6-2, and the 18-year-old Swiss player next faces seventh-seeded Kristina Mladenovic of France.
Dutch player Kiki Bertens saved all five break points in a 6-2, 6-2 upset win against No. 6 Anastasia Pavlyuchenkova of Russia. Next up for Bertens is third-seeded American Coco Vandeweghe.