It's a fixture fit for the final of the competition, with Juve seen as one of the heavy favourites before a ball was kicked after the addition of Cristiano Ronaldo in the summer.
It was Atletico supporters who got the last laugh, though, as a pair of scrappy goals from centre-backs Jose Gimenez and Diego Godin gave the home side a two-goal advantage heading into the second leg in Turin - and the second effort actually deflected in off Ronaldo.
But it was Atletico that had the last laugh as they were 2-0 up with just five minutes to go.
Diego Simeone rang the changes for the final 30 minutes, as Alvaro Morata, Thomas Lemar and Angel Correa were all introduced as Atletico went all out.
Costa spurned a gilt-edged chance to break the deadlock for Atletico five minutes into the second period, pulling his shot wide from inside the area after being played in by Griezmann.
Juve struggled to match the Spanish giants for the majority of the game, with Ronaldo failing to stamp his authority down against his former foes.More news: Actor Jussie Smollett now considered suspect 'for filing a false police report'
The 26-year-old, who was denied by the VAR in the La Liga derby loss to Real Madrid on February 9, headed past Wojciech Szczesny and celebrated passionately but a review found he had pushed Giorgio Chiellini.
Elsewhere in Europe, Manchester City conceded two first-half penalties but rallied to win behind late goals from Leroy Sane and Raheem Sterling.
Ronaldo's blistering free-kick forced Atletico goalkeeper Jan Oblak into a brilliant early save as both sides showed their attacking intentions.
The teams will face off again March 12 in Turin, Italy, in the second leg.
The Argentinian forward said: "I have learnt a great deal form him, particularly in terms of the way we work, how to be a model professional and being fully focused during training so that you know what to expect during games".