Murray struggled with heat as he lost the first set 4-6 to Muller but then produced a sizzling masterclass performance to wrap up the next two 6-1, 6-2.
The 35-year-old Scot, slowed by surgery on both of his hips, must now take on rising ATP Tour powerhitter Jiri Lehecka, aged 21, who also fought back from a set down to beat top seeded Russian Andrey Rublev.
Murray has had a gruelling time in Doha, where he is a two-time winner but a wild card this year.
On Monday, he saved three match points before winning a final-set tie-break against Italian Lorenzo Sonego.
In the second round, Murray was only two points from defeat in the deciding set before surging back to beat Alexander Zverev 7-6 (7/5), 2-6, 7-5.
Against Muller, Murray won the first game without giving away a point and broke his 26-year-old opponent’s first serve, then lost two straight service games as the Frenchman took the first set in 54 minutes.
Murray won a second set in less than 30 minutes and the third in barely 40, as he reached his first semi-final since Stuttgart last June.
There was a dramatic improvement in the veteran’s serve and confidence while Muller’s collapsed.
Czech Lehecka claimed the biggest win of his career in beating world number five Rublev 4-6, 6-4, 6-3.
“The past two months have been very great for me,” said Lehecka, who also reached the Australian Open quarter finals in January. His performances have seen him shoot up to 52 in the world rankings.
Rublev, the Doha winner in 2020, went out in his opening match in his four previous tournaments.
Lehecka said he waited to see his opponent’s tactics before speeding up his play. He is known as one of the tour’s most powerful hitters.
“I just wanted to serve a bit more risky and faster in the second and particularly the third sets,” said Lehecka.