The British third seed was in all sorts of trouble after losing the opening set to Croatia's Marin Cilic and falling 5-1 behind in the second with two service breaks.
However, he rallied to win the set in a tiebreak then rolled through the next two, winning the last 11 games to claim the 3-6 7-6 6-2 6-0 victory.
"I think if you're very angry a set and 5-1 down you're probably not going to be able to get yourself back in the match," Murray said.
"I was more frustrated at the end of the first set when I got myself back into it 4-3 and then lost serve the next game.
"Then I started thinking a little bit more. He got nervous towards the end of that set once I got one of the breaks back, which helped me."
Murray will now play Tomas Berdych for a place in the final, after the Czech sixth seed posted a surprise victory over world number one Roger Federer.
The Scot is still chasing his first Grand Slam title after losing all four finals he has made, including the 2008 US Open and this year's Wimbledon.
But the 25-year-old broke through a major psychological barrier by beating Federer to win the gold medal at the London Olympics and is more confident than ever his major drought could soon end.
He has always had the game to compete with the best but his mental state has been questioned in the past. That is no longer the case.
"For the most part, after the Wimbledon final I did a pretty good job of that as well," he said.
"At the Olympics I just went with the same sort of stuff I was doing at the beginning of the Wimbledon final and sustained it for the whole match. That was why I won."
- Marin Cilic