Karim Benzema and Sergio Ramos looked to have won the tie before Diego Costa scored a late second to equalise.
Saul Niguez's stunning extra-time volley put the Europa League winners ahead before Koke sealed the deal.
Atleti took the lead after just 49 seconds as Costa nodded the ball over Ramos before firing past Keylor Navas from a tight angle.
Julen Lopetegui may have wished he had new signing Thibaut Courtois available as the near-post strike could have been stopped.
But the former Atleti stopper was ineligible as his paperwork was not filed correctly, much to the disappointment of fans on Twitter.
Real drew level as Gareth Bale surged down the right before whipping in a cross on his weaker foot for Karim Benzema to nod home.
Luka Modric – who is being linked with a move to Inter Milan – came on for Marco Asensio after 57 minutes.
And six minutes later, Juanfran handled a cross to hand Los Blancos an opportunity from the penalty spot.
Skipper Ramos stepped up a calmly slotted home from a stuttering run-up as Jan Oblak went the wrong way.
The Champions League winners appeared to be closing out the game until Marcelo gave the ball away near his own area.
Juanfran capitalised on the blunder to find Angel Correa, who set up Costa's tap-in second with 11 minutes to play.
Marcelo had an opportunity to win with the last kick of normal time but he failed to connect properly with an acrobatic effort after Bale found him free in the area.
Atleti took back the lead as Raphael Varane was robbed on the edge of his area and Thomas Partey found Saul to smash a volley into the top corner.
And Koke's low finish from Vitola's cut-back wrapped up victory in a six-goal blockbuster for the fans in Estonia.
It was the first time Real conceded four goals since before Zinedine Zidane took over.
But despite that stat, it will be going forwards Lopetegui is concerned about as his side were contained well for long periods against an admittedly brilliant and well-organised defence.
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