In the semi-finals, the 13th seeded Murray will meet 12th seed Novak Djokovic of Serbia, who overpowered 14th seed David Ferrer of Spain 6-3 6-4.
Murray took a medical timeout in the third game of the second set after rolling his left ankle when diving for a forehand, but he was able to battle past the in-form Haas in their first ever meeting.
"I was in a bit of shock because I've had problems with my ankle before," the 19-year-old Scot told reporters.
"The trainer told me it wasn't too bad and he strapped it and then after 10 minutes it wasn't my ankle hurting, it was my side because I had fallen so hard.
"It was hard to push off on my right leg for my serve and forehand."
Due to his injury, Murray was forced to nudge the ball around the court in the second set, but by doing so, he completely took Haas out of his rhythm.
The creative Murray held on to an early break to take the second set when a frustrated Haas missed an easy backhand return.
Murray began to move better in the third set and broke the German veteran in the third game with a forehand volley winner, but Haas broke back in the eighth with a sublime backhand down the line.
MEDICAL TIMEOUT
Both exhausted, neither man could gain an edge and Haas took a medical timeout for leg cramps after the 11th game.
In a dramatic third set tiebreaker, the 19-year-old Scot fought off his first match point when Haas missed a forehand at 6-5.
Murray then staved off another at 7-6 when he gunned a hard approach shot at Haas that the German could not handle and he took the match when his opponent failed to keep a backhand passing shot in the court.
"It was pretty eventful," Murray said of the two hour, 51 minute contest.
"It was a great match but I was more happy about my ankle not being too bad. There were so many ups and downs.
"I just hope that when I wake up that everything is not too sore because the important thing is my health as I've hurt my ankle before and don't want to risk anything."
Controlling the court with a huge serve, leaping forehand and a solid backhand, Djokovic had few problems with the Spaniard Ferrer, who spent much of the day retrieving.
Despite on court temperatures that soared past 100 degrees Fahrenheit, the 19-year-old Serbian didn't appear to break much of a sweat.
"I felt really good, physically," he said. "I'm playing really some good tennis. That's one of the positive things and it's very important to be physically ready when you're playing on these kind of days."
Second seed Rafael Nadal of Spain and third seed Andy Roddick of the U.S. will contest the other semi-final on Saturday.
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