Three-time Grand Slam champion Murray - now ranked 832 in the world - lost the first set in one hour and six minutes, having taken a 5-0 lead in the tie-break before losing seven points in a row.
Playing in just his third tournament since an 11-month break following surgery, it was an awesome show of resilience from Murray as his hip held out in his third three-set match this week.
However, he recovered to break Copil in the second set.
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.
Now ranked 832nd, Murray ended an 11-month absence from the tour in June, playing just three matches before arriving at the Citi Open.
The three-time Grand Slam victor set up a quarter-final showdown with highly-rated Australian youngster Alex De Minaur.
"I was just grinding every point out, making sure that I capitalized on the points that I didn't do at the French Open", Rubin, an American who lost to Isner in straight sets at Roland Garros, told the media afterward.More news: Martin Devlin: Why the Crusaders will win this year's Super Rugby title
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The No. 2-seeded Isner was coming off a title in Atlanta last week and a semifinal run at Wimbledon last month.
Their all-American matchup was supposed to be played Wednesday but was postponed because of a thunderstorm.
"I basically spent the first match defending, I was playing quite far behind the baseline".
"Finishing matches at 3am is not good". They had met twice before in qualifying matches, most recently in 2014, but never during the real rounds of a tournament. "I was really keen on those big moments, and just had fun with it".
Murray won the final set tie-break 7-4 at 3:02 in the morning before the tears flowed.