Atletico levelled the tie in the 27th minute when midfielder Mario Suarez’s strike found the net via a deflection off Turkey defender Omer Toprak.
Garcia came closest to winning the clash in extra-time before the tie went to penalties, where Hakan Calhanoglu, Toprak and Stefan Kiessling all missed for Leverkusen, while Fernando Torres was on target as Atletico prevailed 3-2.
Substitute keeper Jan Oblak, who has failed to establish himself as Atletico’s No1 since a £12.6m move from Benfica, was an unlikely hero for last season’s Champions League runners-up after replacing first choice Miguel Angel Moya.
The 22-year-old saved Calhanoglu’s spot-kick after Raul Garcia had missed the hosts’ opening penalty, while Leverkusen are yet to reach the last eight of the competiton since 2002.
Two goals from prolific frontman Troy Deeney, meanwhile, fired Watford to the Championship summit following a 2-0 victory over struggling Wigan, a team managed by former Hornets’ boss Malky Mackay.
Brentford consolidated their place in the play-offs courtesy of a 3-2 victory at Blackburn Rovers, with former Luton striker Andre Gray proving the match-winner inside the final 10 minutes.