Murray, 27, beat the American after dropping the first set and facing two match points in the third set.
Murray told Sky Sports News: "It was a big win for me. 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."
The Scot progressed after beating Isner 6-7 6-4 7-6 in America on Thursday.
Murray has won 28 career titles and is currently ranked ninth in the world.