Falling a set behind appeared to spark Murray to life and he grabbed an early break in the second against the world No. 93.
But the Romanian immediately broke back before holding serve to level the match, before Murray's fitness prevailed in the deciding tie-break.
As with the first set, the third saw both players hold their service games through to 5-4 in favour of Murray and the 31-year-old booked his place in the third round on the first break - and match - point after two hours and 34 minutes.
Andy Murray believes the manner of his victory over Kyle Edmund at the Citi Open in Washington was just as important as the result.
The former world number one, who has been sidelined for much of the previous year due to a hip problem, fell a set down against world number 93 Marius Copil.
Manitoba child the province's first human case of West Nile for 2018
The increase in mosquitoes they've identified as being infected with West Nile has come earlier this year than in past years. Previous year in the United States, the West Nile infection was reported in over 2000 patients, causing deaths of 121 people.
The No. 2-seeded Isner was coming off a title in Atlanta last week and a semifinal run at Wimbledon last month.
"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. Play was also delayed Thursday, beginning about 3½ hours late because of a shower. By the end, there were 100 or so fans in the main stadium at the hard-court tune-up for the U.S. Open.
"Finishing matches at three in the morning is not good", he said.
"It's like playing two matches in a day". They had met twice before in qualifying matches, most recently in 2014, but never during the real rounds of a tournament.
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. "I was really keen on those big moments, and just had fun with it". "I don't know how players are expected to recover from that", Murray said.