Muller thundered down 17 aces in beating the Uruguayan and will face two-time defending champion Viktor Troicki in Friday's semis.
Muller thundered down 17 aces in downing the Uruguayan 7-6 (7/5) 6-4 and will face two-time defending champion Viktor Troicki in Friday's semis.
Troicki, on a 13-match winning streak in Sydney, advanced to the last four after German Philipp Kohlschreiber withdrew before their match citing a lower back injury.
Sixth seed Muller knew he would face a tough test against Troicki to make it to Saturday's final.
"He hasn't lost a match in Sydney like, forever. He beat me in the semis two years ago," Muller said.
"I know he's a great player. He's up there for a long time and it's gonna be a tough match."
Muller knew his best chance was to continue serving well as he had done this week in the tournament.
"(I'm) just trying to stay positive, (I'll) try to play well. I think if I serve like I did today it won't be easy to break me.
"I have to focus on that and try to build up from there.
"For me it's the third year in a row for the semi-finals so I'd also like to get one stage further."
It was not always straightforward for Muller, with Cuevas saving six set points before losing the opening set tiebreaker and then Muller winning on his third match point.
"Today was definitely a big step forward. Even though I don't think I played a great match... I played a solid match," Muller said.
"I served pretty well, I was able to focus during the whole match -– there were no ups and downs like in the previous rounds. So I'm very happy about that."
Earlier in the day, German journeyman Kohlschreiber succumbed to lower back soreness that developed a few days ago.
"It's unfortunate. I think I played a good tournament," the 33-year-old said.
"But the improvements are not that big that I'm 100 percent sure to be able to go out on court today.
"It's sad situation. I played some good tennis, and I'm sorry for it."
Russian Andrey Kuznetsov overcame Spanish fourth seed Pablo Carreno Busta 2-6, 6-4, 6-1 and will face either Austrian top seed Dominic Thiem or Britain's Daniel Evans in the other semi-final.