Seven-time champion Federer produced a superb performance on Centre Court and pounced at the key moments to maintain his perfect record in semi-finals at the All England Club with his 10th success at the last-four stage.
The 33-year-old, who is the oldest man to make the final since 39-year-old Ken Rosewall in 1974, will now face defending champion Novak Djokovic in Sunday’s final.
Murray had lost his last three meetings with the world No 2, but he had an early chance to claim a break of serve in the fifth game of the first set.
However, Federer conjured up two strong serves to snuff out the chance and that was the last opportunity Murray had in the opening set as the Swiss enjoyed an 85 per cent success rate with his first serve, hammering down 11 aces.
Murray also had few problems holding his own serve and a tie-break appeared on the cards until Federer seized the moment in the 12th game, claiming the decisive break and the set as he ripped a backhand into Murray’s feet on his second break point.
The statistics were against Murray at this point with Federer having won 93 per cent of his career matches after taking the first set and Federer also looked dominant on court as Murray was forced to save a break point in the fourth game of the second set.
Murray hung in there, though, especially in an exhilarating 10th game which lasted 17 minutes as he saved five set points before finally making it 5-5.
But Federer quickly shrugged off his exertions and took his next service game to love before out-manoeuvring Murray to break the third seed’s serve in the next game and claim a two-set lead.
Murray had fought back to win from two sets down seven times previously in his career, but the third set followed a similar pattern to the first two with few signs of a break of serve in the opening nine games.
And once again it was Federer who stepped up his game when it mattered with the match-winning break in the 10th game as a Murray forehand drifted wide to give Federer the victory in just two hours and seven minutes.
“It’s been tough. Andy has been playing very well for the season,” said Federer, who fired down 20 aces in total, afterwards.
“I’m unbelievably happy. I played so well on the biggest occasion today and that’s probably why I won it.
“I’ve been serving very well for the entire tournament. I kept the pressure up, I went for my shots and was able to mix it up the way I usually do it. It all worked out very well.”
Murray said: “I played pretty well but he served really well. I could have done more returning-wise a few times.
“I don’t know if it was close to his peak, you’d have to ask him, but that was definitely the best he’s served against me. He clearly deserved to win the match.
“At times I played some very good tennis. I served well, probably the best I’ve served in the tournament, so it’s tough.
“I definitely don’t feel I played a bad match.”