The Swiss world number six booked his place in the last four of the tournament on Wednesday as he defeated the Brit in four sets.
Federer won the opening two sets before Murray reduced the deficit by taking the third, but the 17-time Grand Slam champion clinched the fourth to seal his progress.
"Andy played well and put the pressure on me so I'm happy to get the win," he told reporters. "I probably miss more break points than other guys but I kept my composure and kept the poker face and tonight it worked."
Federer will now meet world number one Rafael Nadal on Friday for a place in the final.