The 23-year-old Dimitrov next plays Gilles Muller of Luxembourg, who beat 20th-ranked David Goffin of Belgium 7-6 (5), 6-7 (6), 6-3.
Earlier, Dominic Thiem advanced to the second round with a 6-4, 6-2 victory over sixth-seeded Ernests Gulbis.
The 47th-ranked Thiem broke his Latvian opponent three times and lost just four points on his first serve to claim his first victory this season following consecutive first-round losses.
Seventh-seeded Roberto Bautista Agut also progressed to the second round after defeating German wild card Alexander Zverev 6-4, 6-1.
Simone Bolelli breezed past Lukas Rosol 6-3, 6-1 and will next face either second-seeded Milos Raonic or qualifier Andrey Kuznetsov.