Murray defeated world number 22 Steve Johnson with a with 6-0 4-6 7-6 (7-2) to reach the semi-finals.
Murray, who won with a 6-0 4-6 7-6 (7-2), cruised through the first set, before Johnson levelled after breaking the Murray's serve in the first game of the second set.
Johnson broke again in the third, but Murray rallied and won in a tie-break.
World number two Murray was back out on court later on Friday alongside Heather Watson in the mixed doubles.
The pair are scheduled third on Court Two for their quarter-final with India's Rohan Bopanna and Sania Mirza.