After saving four break points in the first game, the Swede overwhelmed Murray with a dazzling array of deep top-spin forehands, flat backhands and power serving to race through the first set in just 36 minutes.
Although Soderling wobbled when serving for the match at 5-4 and 30-0 up to be broken for the only time, he recovered to clinch the tiebreak 7-4 when Murray netted a backhand.
The Swede punched his right fist into the air in delight and will next face seventh seed Andy Roddick, who powered his way past Spaniard Tommy Robredo 6-3, 7-5 earlier in the day to reach the semi-finals at Indian Wells for a fourth time.
"It's a big win because Andy is definitely one of the best players in the world," Soderling told reporters after reaching his first semi-final in an elite ATP Masters 1000 event.
"I was pretty sure before the match that it is almost impossible to out-power Andy because he moves really well and he puts everything back.
"I really felt it in the first game, which was a crucial hold for me. After that, I started to play not as aggressive and I was waiting a little bit for my opportunities, and it worked pretty well," he added.
Murray, beaten in last year's final by Spaniard Rafa Nadal, was subdued after the match. "He played very well to start with and I didn't," said the Scot.
"I didn't move particularly well. I was hitting the ball really short and he's obviously got a big game and he was able to dictate all of the points.
SERVING WELL
"So from my side it was poor but he hit the ball really big from the back of the court and served well when he needed to. I made some mistakes in the tiebreak, and that's why I lost," he added.
The big-serving Roddick broke the 18th-seeded Robredo in the 11th game of the second set before holding serve to seal victory in 78 minutes.
"I feel like I'm playing really well on my own serve," Roddick said after booking his place in the last four when the Spaniard hit a forehand long.
"I think I've been broken once, but I don't know that I've had that many break points against me throughout the week.
"That's always a good thing in that holding that easily makes my returns better," added the American, whose victory was his 11th in a row against the Spaniard.
Roddick, who won his 28th ATP title in Brisbane in January, outplayed his opponent with his all-round power game, firing down six aces and hitting 19 winners in searing afternoon heat in the California desert.
Nadal and Croatian Ivan Ljubicic reached the semi-finals from the other half of the draw on Thursday.