Two Sergio Ramos headers put Real 2-0 ahead after 20 minutes of the semi-final second leg and Cristiano Ronaldo added a third from a superb break before halftime.
That was his 15th goal of the competition, a record for a single season, and he extended the tally to 16 in the 90th.
Bayern, treble winners last season, were never in the hunt as they slumped to a 5-0 aggregate defeat which will leave a huge stain on coach Pep Guardiola's previously enviable record in the competition.
The only negative for Real was a booking for Xabi Alonso which will rule him out of the final against either city rivals Atletico Madrid or Chelsea.
Real, champions on a record nine occasions, had lost four semi-finals, one quarter-final and six times in the round of sixteen since their last final appearance in 2002.
Bayern, trailing 1-0 from the first leg, laid siege to Real's goal early on while the crowd attempted to intimidate the Spaniards by whistling incessantly when they had possession.
The Bavarians had won eight and drawn one of their previous nine home games against Real and needed a good performance to answer critics who said their possession game had become dull and toothless.
But, although Bayern huffed and puffed, they rarely threatened and there was always a risk they would concede a killer away goal.
That arrived in the 16th minute when Ramos rose unmarked at the far post to score with a powerful downward header.
Amid general disbelief, Real struck again four minutes later when Angel di Maria's free kick was flicked on by Pepe and Ramos headed it into the corner.
Bayern lost composure, leaving them wide open to Real's ruthless counter-attacks.
That was exactly how the third goal came with Real breaking from the edge of their own area to the other end in a matter of seconds, the move ending with Bale feeding Ronaldo to slot past Neuer.
There was an incredible moment shortly afterwards when Neuer came racing out of his goal, and sent his clearance to Ronaldo who tried to score from 40 metres but saw his shot fly over.
Bayern's shaken players got involved in several petulant exchanges as it was clear that their period of European dominance was over.
Guardiola tried to limit the damage after halftime by replacing his only striker Mario Mandzukic with defensive midfielder Javi Martinez.
Arjen Robben produced their best effort with a shot which curled narrowly wide just before the hour but Ronaldo had the last word as Real handed Bayern their heaviest-ever home defeat in European competition.
