Stingy on serve and dominant at the net, the 17-times grand slam champion suffered a wobble that dragged him into a fourth set but recovered his composure to post an impressive 6-3 6-4 6-7(6) 6-3 win under the lights of Rod Laver Arena.
The win continued Federer's revival after his annus horribilis of 2013, and with new coach Stefan Edberg and a bigger racquet in his arsenal, the Swiss master is confident he can give the top seed a shake.
"Physically I know that I can do it. And then because I'm feeling good physically, then I can really think about tactics I want to play, how aggressive or how passive do you want to play," the 32-year-old told reporters.
"I have all these opportunities now. I've been hitting the ball really well for some time now, so it's just nice that it all came together in a big match against Murray like this."
Murray saved two match-points in the third set tiebreak and 13 of 17 breakpoints during the match, but crumbled at 4-3 in the fourth set to concede the decisive break.
Federer made no mistake on his third match point, sealing the thriller with an ace after three hours and 20 minutes.
"I'm explosive out there. I can get to balls. I'm not afraid to go for balls," he said.
"Of course last year at times (I) couldn't do it, but what's important is that I can do it now.
"It was a great game on many levels today, not just physically. Also just mentally it was tough. Then I really played some good tennis. I was very happy."
Having mowed down Jo-Wilfried Tsonga in straight sets in his previous match, the red-hot Swiss appeared set to do the same to Murray, who showed signs of his lack of match practice on the comeback from back surgery.
Federer broke him early and calmly served out the final game of the set to love, sealing it with a huge first serve.
Murray was down a break again at 2-2 in the second set after blasting a forehand crosscourt just wide.
Powerless to breach Federer's serve and outplayed on the baseline, Murray slumped to a two-set deficit after losing a series of attritional rallies.
Pushed to the wall, the Scot saved four break points in his first two service games of the third set, and winced with pain after reaching for a low ball.
TWO MATCH-POINTS
The pressure ultimately told as Murray netted a forehand to be broken at 4-4.
Inexplicably, Federer wavered when serving for the match.
Having not given up a single break point, Federer conceded two with a pair of unforced errors, and Murray sealed the second after running the Swiss ragged.
Dragged into a tiebreak, Federer showcased a brilliant defence to leap to a 5-2 lead and held two match-points when Murray floated a backhand long.
Murray saved both with a gutsy counter-attack, surviving the first with an epic 24-shot rally, before clobbering a forehand down the line to raise set point.
Suddenly tentative, Federer fluffed a half-hearted forehand to concede the set, leaving Murray to punch his fists and roar.
Under siege for 19 minutes in his next service game, Murray saved six break points and then another at 3-2 down.
Federer was unrelenting, however, and had another three chances to break at 4-3. Murray saved the first two, but was beaten by a sumptuous drop-shot that he could only play back to the Swiss who pushed the volley into an open court.
Murray, at 3-5 down, put Federer under pressure by going up 0-30 but the Swiss maestro unleashed three huge serves to kill off the contest, raising thunderous cheers from a sentimental crowd.
Federer will have Swiss company in the last four of a grand slam for the first time, with eighth seed Stanislas Wawrinka playing off the other semi-final against seventh seed Tomas Berdych.
"For me it's an amazing result to again be in the semis," he said following a 2013 season in which he failed to reach a major final for the first time since 2002. "This one feels different because of the tougher times I've had in slams, Wimbledon, at the U.S. Open (last year).
"Plus what I really love is another Swiss in the semis as well ... So that's a big deal."
(Editing by Pritha Sarkar)
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