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Diego Simeone's men were 1-0 down from the first leg but staged a remarkable comeback despite going behind on the night at the Wanda Metropolitano, levelling the aggregate score at 2-2 before triumphing on penalties.
Alvaro Morata had a fine chance after 28 minutes as he was found unmarked in the box but his header was well saved by Yann Sommer. Soon after, Inter opened the scoring on the night after a surgical counter-attack that saw Federico Dimarco played in 12 yards from goal, and he scored with a first-time finish to give his side a two-goal cushion.
Atletico hit back immediately, though, as Inter failed to clear a through ball and Antoine Griezmann finished on the turn.
Marcus Thuram had a gilt-edged chance to put the game beyond Atletico in the second half but he blazed over with 15 minutes remaining. Memphis Depay, on for the ineffective Morata, hit the post in the 85th minute but the ball came back in just seconds later, and the Dutchman made no mistake.
The same could not be said of Rodrigo Riquelme, as he was played in brilliantly by Griezmann in the last minute of injury time and somehow blazed his shot over the bar.
The tie went to extra time, where Lautaro Martinez had perhaps the biggest chance, but he could only send a header wide.
And so it was penalties. Atletico won the shootout in dramatic fashion, as Alexis Sanchez and Davy Klaasen both saw their penalties saved by Jan Oblak, and Martinez blazed the ball over the crossbar to hand the hosts victory. Saul had missed for Atleti, but it mattered little in the end.