The 32-year-old Scot was forced to dig deep against Romanian qualifier Marius Copil, eventually wrapping up a 6-3 6-7 (7) 6-4 victory in two hours and 36 minutes.
Murray is now in position to challenge for his 46th ATP title, although playing three - or potentially four - matches in as many days will be unchartered territory since his return from hip surgery.
The Scot has been feeling his elbow this week as he gets back up to speed with the rigours of serving, and he flexed the arm occasionally.
He could certainly have done with converting one of three points for a double-break in the second set, rather than being taken to a decider.
But the metal hip passed another stern examination, Copil throwing in plenty of drop shots to test Murray's movement in another encouraging step on the road back towards challenging the top.
Three breaks of the booming Copil serve - with one drop of his own - saw Murray wrap up the first set in 44 minutes.
But the dogged Copil, already a break down and seemingly heading for the exit facing further break points, fended them off and hauled himself level at 5-5 before snatching the tie-break, Murray unable to convert his one match point.
The momentum seemed to be with Copli but he blinked first in the decider, two unforced errors and a first double-fault handing Murray two break points.
The two-time Wimbledon champion converted the second and served out, securing victory with an ace, to reach his first semi-final since the French Open in 2017.
Murray told Amazon Prime: "I haven't played a lot of matches in the last few years so when you get to the end of the match it's always difficult to serve it out.
"I played a bad game at 5-3 in the second set and after that he gained a lot of confidence. He served extremely well, he was being a lot more aggressive in the second and third sets.
"Thankfully I managed to get the break at the end, but it was a tough one to get through.
"I feel OK right now, but it's more about how you feel the following day. The good thing about the indoor matches is the points are fairly short, so it doesn't take as much out of you."
