The clash means that Julen Lopetegui got off to a losing start with the Champions League holders.
Madrid had looked set to win the trophy on offer in Estonia in their first competitive fixture following the high profile departures of Cristiano Ronaldo and Zinedine Zidane when a Sergio Ramps penalty put them 2-1 up after the interval.
That came after Karim Benzema cancelled out Diego Costa's first-minute opener, but Atleti fans' favourite Costa hauled Diego Simeone's side level again late on to force extra time.
The Rojiblancos then looked stronger after regular time was up, taking the annual meeting of the Champions League and Europa League winners for the third time in nine seasons.
Real Madrid had landed the title under Zidane in each of the last two years, also emerging victorious against their city neighbours in the Champions League finals of 2014 and 2016.
This result comes after a 2017/18 close season in which Atletico have strengthened while Lopetegui's team is weakened, with no replacement signed for Ronaldo following his transfer to Juventus.