The 2011 champion was almost faultless for most of the first set but the wheels came off in the tie-break as Isner took the initiative.
Murray produced the only break of the match to take the second, but the third went to another tie-break and the eighth seed made his class count as he closed out the match 6-7 (3/7) 6-4 7-6 (7/2).
The duo largely cancelled each other out in the first set, with Isner's big forehand return causing plenty of problems for Murray, but the Scot was easily out-classing his opponent at the net.
Isner was making the early errors, finding the net every time Murray brought him short, but he produced a stunning backhand volley to win the fourth game.
The rest of the set went with serve and Murray secured a mini-break on just the second point before moving 3-1 ahead.
But Isner came roaring back, his big returns proving unplayable off Murray's second serve, and took the tie-break 7-3.
The world number nine could have been forgiven for being shellshocked, but he delivered the perfect response by breaking Isner in the first game of the second set.
Murray had two break points but Isner fought back before eventually succumbing to the first break of the match.
The rest of the set went with serve, with Murray keeping his opponent at bay with little trouble as he closed it out.
That pattern continued until the Briton double-faulted in the 11th game of the third to hand Isner match point, but once again his volleying let him down as he found the net.
Murray saw off another match point at 40-40, but then produced a sublime winner and an ace to take the game to a deciding tie-break.
And the former US Open and Wimbledon champion proved too strong as two good returns forced Isner to find the net. The American 11th seed finished things off with a double fault.
"It was a big win for me," Murray said. "He plays extremely well here.
"Returning his serve, especially his second serve, in these conditions is very challenging so I did well to fight back.
"I felt like I played a very good match. I served well, I moved well, I was hitting the ball clean from the back of the court and when I was challenged at the end of the third set I responded well, that's important.
"It's all about winning right now, I don't really care how I play, I just try and keep getting wins."
Murray was also full of praise for his opponent, whose big serve had caused the 27-year-old problems all match.
"He has the best serve," he added. "He serves huge first serves and the difference between him and the other guys is the second serve.
"He serves quite a lot of second-serve aces, he gets good work on the second serve. I was trying to take it early and even still it's bouncing up round the shoulder. If you take a couple of steps back, it's well above your head."