Following a 1-0 defeat in the first leg in Germany, Atletico levelled the tie in the 27th minute when Mario Suarez's strike deflected in off Omer Toprak.
Raul Garcia went closest to a winner for Atletico in extra time but Leverkusen keeper Bernd Leno saved.
Hakan Calhanoglu, Toprak and Stefan Kiessling all missed for Leverkusen as Atletico won 3-2 in the shootout.
Substitute keeper Jan Oblak was an unlikely hero for Atletico, last year's beaten finalists, after replacing Miguel Angel Moya, who limped off with a hamstring injury midway through the first half.
Oblak has failed to become Atletico's first choice since his £12.6m move from Benfica as a replacement for Thibaut Courtois, who returned to Chelsea after three years on loan.
But the 22-year-old Slovenian saved Calhanoglu's penalty after Garcia had missed the hosts' first kick.
Atletico's Koke also had a spot-kick saved after Toprak missed before Kiessling blasted the 10th and final kick high over the bar, sending the Spaniards into Friday's quarter-final draw along with the already-qualified Real Madrid, Bayern Munich, Paris St-Germain, Porto and Monaco.
It was a cruel finish for Leverkusen, who were hoping to reach the last eight for the first time since since reaching the final in 2002.
They went in search of an early away goal with the lively Karim Bellarabi whipping a low shot wide while a Son Heung-min shot was calmly cleared away by Atletico striker Mario Mandzukic.
Atletico looked tentative at the back without experienced defender Diego Godin, who was suspended, while midfielder Tiago was also banned after being sent off in the first leg. And it seemed to get worse when Moya limped off.
But their fortune changed for the better in the 27th minute when Suarez's low strike flicked off Leverkusen defender Toprak and flashed past Leno, who was unable to react in time.
The Spanish champions almost doubled their lead minutes later with another deflection but Antoine Griezmann's effort clipped Lars Bender to curl wide.
Atletico boss Diego Simeone waved his arms furiously to rally the Vicente Calderon fans at start of the second half and they roared the Rojiblancos forward but they were restricted to half-chances for Arda Turan and Garcia.
Garcia almost won in it in extra time but his drilled shot was parried away by Leno to take the game to a penalty shootout.
Atletico Madrid manager Diego Simeone:
"I loved it. As I watched the match ending I could see we were back again among the best eight teams in Europe with Atletico Madrid in a stadium that was erupting. It left me with goosepimples.
"The fans delight me. The supporters don't realise the strength they give the players."
Atletico Madrid striker Fernando Torres:
"I'm happy, delighted to see the stadium enjoying a match like this. I've often yearned to live nights like this in the Calderon, so it's a day to savour."
Bayer Leverkusen coach Roger Schmidt:
"We are a young team with little experience and it is not easy to hang in there at a stadium like Atletico's.
"In the shootout we lacked concentration and calm. After fighting so hard we are very disappointed. But the players gave their all and so we leave feeling satisfied."
Bayer Leverkusen midfielder Simon Rolfes:
"After a penalty shootout it is difficult to analyse a game. We may not have shown enough courage, but a little luck is also necessary. It is bitter to exit on penalties. Atletico were the more active team and we were not confident enough."
Atletico Madrid reached the quarter-finals of the Champions League after beating Bayer Leverkusen on penalties.