He and Heather Watson then completed a 6-3 6-3 win over Spain's Carla Suarez Navarro and David Ferrer in the mixed doubles.
Murray will face American Steve Johnson in the singles last eight on Friday.
The Wimbledon champion was under pressure going into the third set before winning six games in a row to progress, in the process stretching his winning run to 15 matches following his French Open final defeat by Novak Djokovic in June.
"It was really, really hard out there and difficult to appreciate unless you were actually out there playing," Murray said.
"Sometimes the conditions don't allow for you to play really well and the conditions didn't allow that today."
"It was really, really hard out there and difficult to appreciate unless you were actually out there playing," Murray said.
"Sometimes the conditions don't allow for you to play really well and the conditions didn't allow that today."
It was a different matter in the doubles, where he and Watson competed a speedy victory after they received a late call-up when Romanians Monica Niculescu and Florin Mergea withdrew from the competition.
Watson, who was knocked out of the second round of the women's singles, admitted she "had her bags packed" before hearing she was playing.
Watson, who was knocked out of the second round of the women's singles, admitted she "had her bags packed" before hearing she was playing.
