Murray, who improved his career record against Federer to 5-2 in tour matches, will face Andy Roddick in Saturday's final.
"It is good to have beaten him (Federer),'' Murray said. "He is an awesome player, arguably the greatest player of all time. So, I am happy about my performance today. ... The first set was very tight as the tiebreak was bit of a back-and-forth thing for both of us. I lost that and all of a sudden there was a lot of pressure on me.''
Since losing to Federer in the U.S. Open final, the fourth-ranked Murray beat the Swiss star at the Madrid Masters in October, the Masters Cup in Shanghai in November and an exhibition tournament in Abu Dhabi last week.
"It is disappointing to lose after the first set when things were going my way,'' Federer said. "I just couldn't give the knockout punch. But it is not a big thing.''
"Andy is a tough player against me,'' he added. "I hope when big matches come, I beat him.''
Federer has 13 Grand Slam titles, one short of Pete Sampras' record, heading into the Australian Open later this month.
On Friday, Federer committed four double faults and had 37 unforced errors, the same number as winners. On match point, Federer slammed an easy smash into the net.
While Federer failed to break serve, Murray converted four of six break points.
Murray took an injury time-out in the third set for treatment on his lower back, but showed no signs of trouble as he closed out the match.
Earlier on Friday, Roddick reached the final by defeating Gael Monfils 7-6 (1), 3-6, 6-3.
After slumping in the second set, Roddick broke his French opponent twice in the third to secure the win in 2 hours, 22 minutes.
"It was a very tough match,'' Roddick said. "He moves effortlessly and can win some tough and long points. ... He was using a lot of drop shots today. He mixes his game well.''
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