The Brit fought back from two sets down inside Rod Laver Arena on Wednesday, but it was not enough as Federer went on to close out the match 6-3 6-4 6-7(6) 6-3.
However, Murray admitted that he is pleased with his progress at the tournament after recently recovering from back surgery.
"I was proud of the way I fought," the world number four told reporters.
"I changed my tactics a little bit, started playing a little bit more aggressive, and that was maybe my undoing a little bit at the end, because I really started going for my shots to get myself back into the match."
Federer's reward for beating Murray is a semi-final encounter with old rival Rafael Nadal.