The Scot will play Bernard Tomic next after the Australian beat Japan's Kei Nishikori 7-6 (7-1) 7-6 (7-5).
In the women's draw, Murray's fellow Briton Johanna Konta lost 6-7 (1-7) 6-4 6-0 to fifth seed Agnieszka Radwanska.
Germany's Angelique Kerber reached the last eight with a 7-6 (7-5) 6-4 victory over Barbora Strycova and will play Carla Suarez Navarro.
Kerber will replace Serena Williams as world number one if she wins the Cincinnati title. Williams missed the tournament with a shoulder injury,
