The 31-year-old lost his first Grand Slam encounter against the Scot 6-4 6-7(4) 6-3 6-7(2) 6-2 in four hours at the Rod Laver Arena.
Federer created just six break points during the five-set contest and admitted that he found it difficult to break down Murray's serve.
"I think overall he probably created more chances than I did," said Federer. "I had difficulties getting into his service games time and time again.
"I think he started off serving well, and then, I mean, fifth set, obviously he did well. I think he played a bit more aggressive because he did create more opportunities over and over again.
"I think it was a tough match. I think I had my chances a little bit. Obviously you're going to go through a five setter with some regrets, you know. But overall, like I mentioned, I think Andy was a bit better than I was tonight."
Murray will now play Novak Djokovic in the final.