The pair shared breaks in the third set, and as the clock ticked on later into the night, Murray eventually emerged victorious (7-4) in the final tie break - at 3:01 a.m.
His quarterfinal opponent will be Australia's Alex de Minaur, who beat eighth-seeded Hyeon Chung of South Korea 6-2, 4-6, 6-2 in the third round.
But speaking to a small group of reporters outside the locker room, Murray said he "potentially" could consider not playing.
'I don't think it's reasonable.
"Just the emotions coming out at the end of an extremely long day", Murray said, "and a long match".
"When you're expected to come back and perform the next day, I think that's unreasonable". The competitive spirit he has displayed in his first three matches in Washington are indicative of just how much he loves to compete. It's not good for the players. It's not good for fans, TV. By the time you're done with all your recovery and stuff, it's going to be 5.30, 6 o'clock in the morning.
He added: "There are lots of positives to take from this week, so I'll take some time to rest and recover - I won't play in Toronto next week - and then head to Cincinnati early to prepare and get ready". It's not good. It's basically like playing two matches in a day'.
His task was even tougher here after Romania's Copil fought back from 5-0 down in the first-set tie-break to win it 7-5.
Murray, who continues his return from hip surgery, overcame the Romanian 6-7 6-3 7-6 in his second three-set encounter in two days, having previously defeated fellow Briton Kyle Edmund.
But the former world No 1 quickly regained his composure with an immediate break of serve at the start of the second set before a second break of the match ensured the match would go the distance.
Another breaker was required to decide the match, yet Copil's backhand into the net handed Murray victory after a three-hour marathon.
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