Atletico spurned the chance to win the match when Leverkusen goalkeeper Lukas Hradecky saved Yannick Carrasco's spot kick before Saul Niguez's follow-up strike hit the crossbar.
The result ensured Porto go into next February's knockout stage, with the Portuguese side on nine points in group B after a 4-0 win at leaders Club Brugge, already guaranteed a top-two finish on 10 points, earlier on Wednesday.
Atletico, without a win and goalless in their previous three Champions League outings, exit the competition at the group stage for only the second time in the past 10 years.
Leverkusen forward Adam Hlozek set up Moussa Diaby, who scored in September's 2-0 home win over Atletico, for the France winger to blast a shot from a tight angle into the far top corner and put the visitors ahead in the ninth minute.
Atletico levelled in the 22nd through Belgian Carrasco's low shot inside the near post from the edge of the box.
But determined Leverkusen went back in front seven minutes later when on-loan winger Callum Hudson-Odoi fired a shot down the middle past goalkeeper Jan Oblak.
Atletico once more came from behind to equalise five minutes into the second half through an impressive goal from substitute midfielder Rodrigo De Paul before their last-gasp penalty miss.
Atletico travel to Porto in their last group stage clash on Tuesday when Leverkusen host leaders Brugge as the Spaniards and Germans fight it out for a Europa League knockout spot.