The British world number two, 28, showed excellent resilience in the face of the American's powerful serving to win 6-7 (4-7) 6-4 6-4.
Murray got the decisive break in the seventh game of the final set before serving out to seal victory after two hours and 31 minutes in China.
He will face Czech Tomas Berdych in the last eight today.
"Your patience is certainly tested when you play against guys like John," Murray told Sky Sports.
"It's always tough against him - you can feel like you have control when you are into rallies but it's tough to get into them sometimes,"
Murray has won all five meetings with the 30-year-old 13th seed.
