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 on Friday for a place in the last four.
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.