After a 6-7 6-3 7-6 win, which included a final set tie-break and did not finish until after 3am local time, the emotional rollercoaster caught up with Murray.
The 31-year-old lost the opening set 7-5 on a tie-break, but fought back strongly to level up the match with a commanding break of serve in the second at a crucial stage.
Andy Murray, a former number one and three-time grand slam champion, later hit out at the schedule.
Thursday's play at the hard-court tuneup for the U.S. Open was delayed at the outset by about 3 1/2 hours because of rain.
He explained: 'Finishing matches at three in the morning is not good. It's not good for anyone, I don't think, involved in the event.
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At 3:01am in Washington, after grinding out yet another remarkable victory to reach the quarter-finals, Andy Murray sat down in his courtside chair and sobbed uncontrollably into his towel for more than two minutes.
Murray will play Alex de Minaur of Australia in the quarter-finals - his first since Wimbledon in 2017.
Speaking about his condition after the Copil match, Murray said: "It doesn't feel great, just now. I don't know how players are expected to recover from that", Murray said. By the time you're done with all your recovery and stuff, it's going to be five thirty, six o'clock in the morning. It's not good. It's basically like playing two matches in a day'.
Copil proved a testing opponent, breaking back when Murray appeared set to move clear in the decider.