Murray has struggled for form and fitness since soaring to the top of the world rankings at the end of last year.
However, the Scot had far too much guile for the big-serving world number 104 as he didn't face a single break point in sealing victory in just under 90 minutes on court.
"I still think there's a ways to go. I don't think I'm playing perfectly just now," said Murray. "I had no break points against me. That was something that obviously I'd struggled with, especially the last couple of tournaments.
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"It's a good sign for me that my serve is getting better. Hopefully that continues that way. "When I serve well, the rest of my game tends to follow. It's an important part of the game for me."
Murray will face either Frenchman Pierre-Hugues Herbert or Croatia's Borna Coric in the last 16 on Thursday.
The Wimbledon champion passed up two chances to break the Copil serve in the second game of the match.
But Murray eventually made his move and let out a cry of joy with a brilliant cross-court half-volley to break Copil's resistance and take the first set 6-4.
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One break in the sixth game was also enough for Murray to take the second set as he easily served out for the match.
Murray's conqueror in Barcelona last month and potential quarter-final opponent Dominic Thiem eased past American Jared Donaldson 6-3, 6-4.
Ninth seed David Goffin became the first man into the third round with a routine 7-6 (7/3), 6-0 win over Germany's Florian Mayer.
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