The top seed came through 3-6 7-6 (7-2) 6-4 at the Aegon Championships.
Muller, ranked 48th, played clinically until the second set tie-break, when Murray stepped up a gear.
The British number one goes on to play Viktor Troicki after the Serb beat John Isner 7-6 (7-5) 6-3 in the day’s last quarter-final.
Saturday’s other semi-final will see South African Kevin Anderson take on France’s Gilles Simon.
Murray, 28, eventually found his sharpest footwork, and with it a way past the big-serving Muller, after struggling to make much impression for the first 70 minutes.
“I started to read the serve a bit better at the end of the tie-break and that was when the match changed,” the Scot told BBC Sport.
“It’s always tough when you’re behind and I wasn’t getting many chances, but at the end I started to return better and relaxed a bit more, and I played some good tennis.
“Hopefully I can take that form into the semi-finals.”
Muller is coached by Briton Jamie Delgado, Murray’s friend and former Davis Cup team-mate, and for the best part of two sets the outsiders looked to have devised a winning strategy.
The 32-year-old grabbed his chance to break for a 3-1 lead and held on under pressure in the next game with some calm serving.
Muller moved smoothly towards the finish line for the next hour or so, seeing out the first set in 32 minutes and denying Murray another break point as the second progressed.
Murray was lambasting his own movement and, urged on by coach Jonas Bjorkman on the sidelines, the Scot finally made his move in the tie-break.
A running forehand pass had Murray’s supporters on their feet and he then attacked a second serve for the double break, taking the tie-break for the loss of just two points.
Muller’s cool exterior finally cracked and a first-serve percentage hovering around 30% left him vulnerable early in the decider.
Murray was now bouncing around the baseline, eager to attack the Muller serve, and a thumping forehand down the line brought him what proved to be the decisive break.
For all Muller’s early excellence he had been unable to get anywhere near the 37 aces of his first-round match, and by the end had managed just eight to Murray’s 13.
A beautiful Murray forehand lob and more winning drop shots saw Muller pulled in every direction, and the Briton sealed victory after one hour and 56 minutes.
He will play Troicki for the seventh time on Saturday and has a 6-0 record, but they have only met once since the Serb returned from a 12-month ban last July for failing to take a doping test – something Murray described at the time as “unprofessional”.
“Obviously you have to be aware of what exactly the rules are,” Murray said on Friday.
“Like I said, it’s extremely serious. I think if you want to protect the image of your sport, you need to understand how serious the drug testing is and anti-doping is.
“I stand by what I said there. I’m sure he’s learned a lot from that.”
In the earlier quarter-finals, Anderson beat Guillermo Garcia-Lopez 7-6 (9-7) 7-5, and Simon converted his fifth match point to edge out Milos Raonic 4-6 6-3 7-5.