Bale, signed for a world record £86million from Tottenham last summer, indelibly wrote his name into Real folklore with the goal that took the team towards 'La Decima' - the club's 10th European Cup triumph.
The defeat and final scoreline was harsh on city rivals Atletico, who had been seconds away from their first-ever success in the continent's biggest club competition after Diego Godin's 36th minute opener, only for Sergio Ramos to power home a headed equaliser in the third minute of added time.
Real's victory means coach Carlo Ancelotti joins legendary Liverpool boss Bob Paisley as the only men to win three European Cups, while Bale again proved the man for the big occasion having also led his new club to Copa del Rey glory earlier this season.
With Atletico tiring and time running out in extra time, substitute Marcelo was allowed to advance unchecked and fire under Thibaut Courtois, before Cristiano Ronaldo added further gloss from the penalty spot after being brought down by Juanfran.
Atletico's gamble on the fitness of star striker Diego Costa proved flawed with only nine minutes gone when the Spain international was forced off by the hamstring injury he sustained just a week ago against Barcelona.
Raul Garcia became the first player to go into referee Bjorn Kuipers' book for a cynical foul on Angel Di Maria, with Ramos cautioned for remonstrating with his rival and Courtois called into action to gather Ronaldo's free-kick as the wall in front of him parted.
Bale missed a good opportunity to break the deadlock, seizing on a terrible square pass from Tiago and bearing down on the Atletico box, only to scuff his shot wide as Miranda threw himself into a last-ditch challenge.
Atletico took the lead in the 36th minute as Iker Casillas' foolish foray off his line proved costly as Gabi's corner was headed back into box for Godin to rise above Sami Khedira and nod home, despite the frantic scrambling back of the Real goalkeeper.
Miranda went into the book for pulling back Di Maria in the 52nd minute and Courtois had to make his first serious save of the contest as he tipped Ronaldo's deflected free-kick over the bar.
Adrian Lopez, who had been sent on in place of Diego Costa, saw his shot deflected wide, before Ancelotti elected to make a double change and sent on Isco and Marcelo in place of Khedira and Coentrao.
Ramos joined a Real attack and swung a delightful cross into the box which Ronaldo looked destined to head home, only for the ball to flick off his forehead away from goal and out of reach of Karim Benzema at the far post.
There was a nasty collision between David Villa and Casillas as the Atletico man slid in on a dangerous low cross, with the striker harshly booked, before Bale smashed a shot wide of Courtois' goal from the edge of the box.
Bale had another opportunity 13 minutes from time as he burst away from Godin and cut in, but he again fired wide, with Godin then forced into a last-ditch lunge to deny Isco as Atletico were forced further back.
The closing stages were played out almost entirely in Atletico's final third as Real poured forward, and just when it looked as though Diego Simeone's superbly drilled La Liga winners would hold on, Ramos rose to power home a header.
Given the relentless nature of the preceding 90 minutes, it was no surprise that the pace dipped somewhat in extra time, before Bale capped his debut season with Real with the biggest goal of his career in the 110th minute.
Marcelo (118) and Ronaldo (120+1) rubbed salt into Atletico's wounds, with coach Simeone losing his head in the closing seconds and venturing onto the pitch to confront Raphael Varane before being sent from the field as the final whistle blew.