Federer raced into a two-set lead but failed to serve for the match at 5-4 in the third and threw away two match points in the subsequent tie-break, before eventually closing out the contest 6-3, 6-4, 6-7, 6-3.
But Murray, playing in just his second tournament since undergoing back surgery last September, feels he is close to rediscovering his best tennis after encouraging performances at the Rod Laver Arena.
“I don’t know how many players have come back from surgery and won the first grand slam back, in their second tournament. It’s very unlikely to happen,” he said.
“I just need to use this as a stepping stone to getting better and be happy that I’ve got through five matches – the last two were particularly tough.
“I’m playing at a decent level fairly quickly again. Hopefully I’ll be back playing my best tennis soon.”
Federer will play Rafael Nadal in tomorrow’s semi-final, after the world No1 saw off Grigor Dimitrov 3-6, 7-6, 7-6, 6-2.
Meanwhile, No2 seed Victoria Azarenka’s charge toward a third straight title in Melbourne was ended by Agnieszka Radwanska 6-4, 5-7, 6-0.