Muller thundered down 17 aces in ousting 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.
Austrian top seed Dominic Thiem went out to British number three Daniel Evans 3-6, 6-4, 6-1 in just over two hours in the evening match.
It was the 26-year-old Briton's first win over a top-10 ranked player and it will be his second ATP World Tour semi-final since Zagreb in 2014.
Lleyton Hewitt and Pete Sampras have won back-to-back titles in Sydney but no-one has lifted the Sydney trophy in three consecutive years since Australian John Bromwich from 1937-40.
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
- Golden run -
Troicki continues his golden run in Sydney and is shaping up for another final.
"It's pretty weird to go into the semi-finals with only one win," Troicki said, having enjoyed a bye in the opening round.
"This is by far the most-successful tournament for me ... I also played one final here in 2011," he said.
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.
Russian Andrey Kuznetsov overcame Spanish fourth seed Pablo Carreno Busta 2-6, 6-4, 6-1 and will face the 67th-ranked Evans in the other semi-final.
"That's pretty much the best win of my career," Evans said of his upset of Thiem. "In the end he didn't really know how to get it by me."