The late-blooming 34-year-old chalked up a seventh successive victory on the green stuff as he beat the powerful American 6-4 7-6(5) to reach the semi-finals.
Next in Muller’s firing line will be fourth seed Marin Cilic, one of three former champions to reach the last eight, who was equally impressive in dispatching American Donald Young 6-4 7-5 on another sunny day at the prestigious London club.
Grigor Dimitrov, who won the title in 2014, also made it through after a mid-match hiccup against young Russian Daniil Medvedev who needed shoulder treatment after appearing to injure himself while practising his smashing.
Bulgarian Dimitrov, seeded six, won the opening five games but made life difficult for himself before winning 6-3 3-6 6-3.
Medvedev, ranked 60, was a break up in the decider but Dimitrov hit back to win with something to spare.
He will face Spain’s 35-year-old Feliciano Lopez, who beat seventh seed Tomas Berdych 7-6(5) 6-7(1) 7-5, in Saturday’s semi-finals.
Muller had never won an ATP title before this year.
He opened the year in style by putting that statistic to bed in Sydney, and last week ruled on the grasscourts of Den Bosch where he beat serving demon Ivo Karlovic in the final.
Now, after beating Querrey, he is closing in on a third.
There was little between them and the match boiled down to one poor service game by Querrey early in the first set when Muller broke to love with a beautifully angled winner after chasing down a drop volley by the American.
There were no breaks in the second set although Muller had to fend off two set points at 5-6, the first after turning around a rally Querrey had been in control of with a crafty lob and the second with an unreturnable serve.
Muller, who cracked down 12 aces with a swinging serve that is tough to read, finished the job with a feathered drop volley.
“I’m going to play semi-finals here for the first time tomorrow. I lost in the quarter-finals for the two years, so I’m really happy that I was able to go one step further,” he said.
Muller confirmed he had pulled out of next week’s Eastbourne tournament to rest up ahead of Wimbledon.
(Editing by Ed Osmond)