Murray struggled early on with his strapped-up thigh, and was flummoxed in that same stage by the craft and guile of Muller’s serve, but rallied for a 3-6 7-6 (7/2) 6-4 triumph.
Should the heavy rain forecast wash out Saturday’s schedule, however, Murray could end up having to play both his semi-final and the final on Sunday to claim the title.
At this stage the 28-year-old Scot will not be overly perturbed, especially given the fact he had to free himself from the choke-hold of Muller’s serve that propelled his opponent into a one-set lead.
The 32-year-old, coached by Murray’s fellow Brit and close friend Jamie Delgado, was unable to sustain that howitzer start, however, slipping off the pace and out of the tournament.
Three-time Queen’s Club champion Murray steadied himself after his shaky opening, eventually devising a way to dismantle Muller’s serve en route to the semi-finals.
Murray will face the winner of the final last-eight tussle between Viktor Troicki and John Isner in a Queen’s week mostly dominated by the big servers.
Chelsea manager Jose Mourinho watched on as Murray edged ever closer to that fourth Queen’s title that would see him equal the existing record haul.
Hot weather and breezy days have meant for hard, fast courts at Queen’s this week, favouring those with the siege-gun serves.
Murray actually sent down 13 aces to Muller’s eight, but his opponent took his tournament tally to 64, underlining his chief weapon’s potency.
Muller’s variation and accuracy kept Murray subdued at the outset, but the Scot pulled off an impressive deconstruction to secure victory.
Should Wimbledon follow Queen’s power-serve suit, Murray has shown himself well able to cope.
Muller put Murray on the back foot from the off, converting the first break point in the match to lead 3-1.
The 32-year-old then underlined his tenacity by fighting off two immediate break points to hold and cement control of the set.
Given power serving has been the theme at Queen’s so far this week, it was wholly appropriate that Muller completed that vital hold with an ace.
Murray saved a break point in the next game before producing a lovely slice drop shot to hold serve and hint at a comeback.
But Muller seized the chance when serving for the set, converting the first of three set points by holding to love.
The second set stayed with serve throughout, Murray again unable to gain any headway when receiving.
Once it came to the tie-break though, the two-time grand slam champion’s class finally started to tell. Murray eased through 7/2 to claim the set and level the match.
Finally in the groove, Murray struck first blood in the final set, converting the second of two break points for an immediate lead.
The 28-year-old had the read he required on Muller’s otherwise-bewitching serve and did not look back thereafter.
Muller saved a match point with that trusty serve but could ultimately not recover, leaving Murray to serve out the match.