Fritz was not the only player pulling double duty at the combined ATP and WTA hardcourt tennis tournament, after rain brought play to a premature end on Thursday.
Britain's Daniel Evans defeated Alexander Shevchenko 6-4, 6-3 then came back to upset second-seeded hometown hero Frances Tiafoe 6-4, 7-5.
Tiafoe had made short work of his third-round opponent, dispatching Chinese teenager Shang Juncheng 6-2, 6-3.
But ninth-seeded Evans gained an early break in each set and turned back the American's bid to rally in the second, breaking Tiafoe for a 6-5 lead and saving a break point as he served out the match in the next game.
Evans next faces Bulgarian fifth-seed Grigor Dimitrov, who got some unexpected rest when France's Ugo Humbert withdrew from their quarter-final with a leg injury.
World number nine Fritz, coming off his sixth career ATP title last week in Atlanta, rallied to dispatch Murray after a three-hour battle.
Murray, showing his formidable range of skills at age 36, won his first break chance of the match with a forehand cross-court drop volley winner on the line, breaking Fritz and pull level 5-5 on the way to a first set tie-break.
Murray smashed a forehand winner for a 3-1 lead, took a quick break to replace a shattered shoe, then took two points off Fritz's serve for a 5-2 lead. He closed out the first set two points later on a forehand winner.
Fritz battled back to take the second and they fought to 4-4 in the third before Murray swatted a backhand wide to surrender a break.
Murray roared to a 40-0 lead in the 10th game but Fritz saved three break points, the last on his 17th ace. Murray denied him on two match points but Fritz blasted a forehand winner on his third chance for the victory.
The 25-year-old American's reward was a later quarter-final against Australian Jordan Thompson, who ousted American Christopher Eubanks 6-2, 6-2 in another match postponed from Thursday because of rain.
Fritz would also ease past Thompson, winning 6-3, 6-3.
On the women's side at the ATP and WTA hardcourt event, top seed Jessica Pegula outlasted Wimbledon semi-finalist Elina Svitolina 4-6, 6-3, 6-4 to reach the semi-finals.
The American will face Greek fourth seed Maria Sakkari, who beat Canadian Leylah Fernandez 7-5, 6-2 on Friday in a rain-postponed match before topping Madison Keys 6-3, 6-3.
Third-seeded American Coco Gauff cruised past sixth-seeded Swiss Belinda Bencic 6-1, 6-2 to line up a semi-final against defending champion Liudmila Samsonova, a 6-4, 6-2 winner over Ukraine's Marta Kostyuk.
Gauff, in contrast, was never troubled by reigning Olympic champion Bencic, converting six of her nine break chances.
"I put a lot of hours in after Wimbledon and I was eager to bounce back," said Gauff, who crashed out in the first round at the All England Club.
"The serve has improved, the return has improved and I think I'm dictating a lot of points, which against Belinda is not an easy thing to do."