Murray couldn't convert any of his first seven break points but finally broke Cuevas in the final game of the first set. He then saved four break points in the opening game of the second before breaking again for a 4-2 lead.
The former No. 1 is now 7-7 on tour since returning to action after having hip surgery in January. He won his first singles match at a tour-level event in September at Zhuhai, China, then made it to the China Open quarterfinals in Beijing this month.
''I felt a bit better today. I thought I served quite well for most of the match and when I was able to get into the baseline rallies, I felt I was hitting the ball quite clean, which is good,'' Murray said. ''It's getting there. It sometimes takes me a little bit longer to get warmed up in the matches and sort of feel confident with my movement. But today I think from the beginning of the match I started pretty well. It was a good day.''
Murray will next face Romanian qualifier Marius Copil, who upset third-seeded Diego Schwartzman of Argentine 6-4, 5-7, 7-6 (7).
American Frances Tiafoe advanced by beating seventh-seeded Jan-Lennard Struff 6-3, 6-4.
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