Murray had never beaten the Swiss in a Grand Slam tournament - with the Scot losing in the final of the 2008 US Open, the 2010 Australian Open and last year at Wimbledon - but he gained some revenge by winning in exactly four hours.
He will meet top seed and defending champion Novak Djokovic in Sunday's final after the Serbian sent out a devastating message that he was in top form with a 6-2 6-2 6-1 demolition of fourth seed David Ferrer on Thursday.
A solitary break of serve in the third game saw Murray take the opening stanza in 45 minutes but, despite only facing one break point on his own serve in the opening two sets, the third seed found himself level.
Murray had been barely tested in Melbourne, wrapping up his previous five matches in straight sets and spending less than nine hours on court, but he lost the second set tie-break 7-5.
However Federer, who was bidding to reach his 25th Grand Slam final, was broken in the sixth game of the third set and the 25-year-old Scot continued to serve impeccably to take a two sets to one lead with his 16th ace of the match.
The 31-year-old Federer had been seriously tested in a 3-1/2 hour five-set quarter-final against seventh seed Jo-Wilfried Tsonga, but showed typical grit to save a break point in the opening game of the fourth set before breaking for the first time in the match to lead 3-1.
But Murray, a player who is on the cusp of overtaking the Swiss as the second best in the world, broke back in the seventh game and levelled at 4-4 after surviving a break point.
The Olympic champion then crucially broke the 17-time Grand Slam champion in the 11th game but Murray failed to serve out for the match at 6-5 and lost the resulting tie-breaker 7-2 as Federer levelled the match once more after a 71-minute set.
However Murray would not be denied and broke Federer at the first opportunity in the deciding set to roar into a 3-0 lead before the Swiss hit a forehand long in the eighth game to allow his opponent to reach his sixth Grand Slam final.
In the final set, where once he might have run through a mentally-scarred opponent, Federer ran out of steam and in the end, was well beaten.
"Andy was just a bit better today," Federer said. "I think overall he created more chances than I did.
"I struggled to get into his service games time and again the way I usually do and then in the fifth set, he did well and was more aggressive.
"I was down in the score, basically from the start.
"Definitely it (felt) more like a chase. I was able to level it a couple of times.
"I think it was a tough match. I think I had my chances a little bit. But Andy was a bit better than I was tonight."
Murray, who ended Britain's 76-year wait for a male Grand Slam winner at the US Open, will now try to become the first man in the professional era to win his second Grand Slam immediately after his first against Djokovic in a repeat of September's Flushing Meadows final.
"I didn't see much of Djokovic's semi-final but I heard he played extremely well. This is his third final in a row here and so he plays great here and I will have to play my best to win," Murray said in a courtside interview.
"I am sure if I get myself in a position to win, it may be easier than it was at Wimbledon for example. Every time we play each other it is a physical match, he is an unbelievable mover so I will need to be ready for the pain, but I hope it is a painful match because that means it will be a good one."