Having been 3-0 down from the first encounter, Cristiano Ronaldo extinguished any faint hopes of a comeback from Atletico with two well-taken early penalties.
The Portugal international's seventh-minute opener made life almost impossible for the hosts before he completed his brace just nine minutes later with another spot-kick to hand the visitors a dream start to the second leg.
Ronaldo was unable to complete his hat-trick, however, as the 18-time winners booked a final clash with either Barcelona or Real Sociedad thanks to a 5-0 aggregate win.
Real also created a little slice of history by becoming the only team to reach the Copa del Rey final without conceding a goal.
Gareth Bale threatened first, glancing a header wide from Xabi Alonso's delivery, before Ronaldo was brought down in the area by Javier Manquillo after leaving Miranda for dead with an electric turn of pace.
Raul Garcia struck Real's post with an impressive long-range effort five minutes later, but Diego Simeone's men were hit with another blow when Bale was fouled by Emiliano Insua in the area.
Referee Alberto Undiano again pointed to the spot and Ronaldo firmly placed the ball in the same corner as before to double Real's advantage in the game after just 16 minutes.
The home side's best spell came in the final 10 minutes of the half when Garcia almost found himself through on goal, only for Varane to clear, before Cristian Rodriguez fired over from the left-hand edge of the box.
Opportunities were limited in the early parts of the second half, with Bale's dipping free-kick causing Aranzubia problems 25 minutes from time.
Atletico continued to look neat in possession but failed to get behind Real other than when Jose Sosa rifled an effort at Iker Casillas - forcing the Spaniard into a smart save.
Defeat represents a third straight loss for Atletico without finding the net, with the sides due to renew rivalries in La Liga in March.
