Murray, a three-time Grand Slam champion, is attempting to work his way back to form almost 10 months after career-saving hip surgery.
The Scot, currently ranked 243rd in the world, held serve throughout against eighth seed Cuevas before sealing victory in 84 minutes.
He will take on Romania's Marius Copil -- who hit 13 aces past third seed Diego Schwartzman in a 6-4, 5-7, 7-6 (9/7) win -- on Friday for a place in the last four.
"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.
"He served very well, which made it difficult for me to get the breaks, but I thought for me it was a good match."
Italian teenager Jannik Sinner dumped out top seed and 2018 runner-up Gael Monfils 6-3, 6-2 to reach his first ATP quarter-final.
Next up for the 18-year-old Sinner is American Frances Tiafoe, a straight-sets winner over seventh-seeded German Jan-Lennard Struff.
"I played a very solid game. He didn't play his best tennis for sure. It's not easy if you make the final the year before and then to come here to play a young player and wild card," world number 119 Sinner told atptour.com
"I think I played solid today. I served good on the important points and I think that was the key today."