Oblak produced a remarkable triple save midway through the second half as Atletico held on to their 4-2 first-leg advantage to take their place in Friday's draw for the last eight.
"It is not easy to be in the quarter-finals four years in a row," Oblak told BeIN Sports Spain. "I hope Atletico will continue like this, keep growing like in the last few years and I am sure we can achieve great things."
On a night of few Atletico chances, the home fans didn't even get to cheer the return of hero Fernando Torres.
Atletico eye Griezmann inspiration for Leverkusen test
Two weeks on from being knocked unconscious in a sickening head clash Atletico boss Diego Simeone left Torres on the bench for the full 90 minutes.
"There is always a feeling of enthusiasm, passion, the Champions League is always special," said Simeone, who has led Atletico to the final in two of the past three seasons.
"Evidently, this is a very important step for the club."
New Leverkusen boss Tayfun Korkut was taking charge of his first Champions League game after Roger Schmidt was sacked when Leverkusen's first-leg defeat was followed by a 6-2 thrashing at Borussia Dortmund.
Korkut summoned a much-improved performance from the Germans as they forced Atletico onto the back foot for long spells.
Torres cleared of any brain injury
However, the damage had already been done as the hosts registered their 17th clean sheet in their last 21 Champions League home games.
"I congratulate Atletico Madrid and especially their 'keeper Oblak for a wonderful performance," said Korkut. "I haven't seen anything like [the triple save]. I've watched it again and it isn't just him. There were many defenders throwing themselves to block the shot. You see how this team defends their goal."
Leverkusen had the chance to bring the tie back to life after just four minutes but Kevin Volland pulled his shot across Oblak and beyond the far post.
Atletico took until seven minutes before the break to threaten but then forced Bernd Leno into two fine saves in 60 seconds.
Firstly, Angel Correa was denied before Koke's low driven shot was turned behind by the German international.
Leverkusen look to extend unbeaten streak against Atletico
Only a poor cross from Volland denied Javier Hernandez what should have been a tap-in at the start of the second half after great work from Julian Brandt.
At the other end, Atletico went close twice in quick succession once more as Correa danced inside two Leverkusen challenges before stabbing just wide.
Antoine Griezmann was then inches away from his 22nd goal of the season with a measured chip over Leno that floated just wide.
The highlight of the match came 21 minutes from time when Oblak pulled off an incredible triple save to end Leverkusen's hopes of a comeback.
Brandt pounced on a slack pass from Jose Maria Gimenez and after he was denied one-on-one with Oblak, the Slovenian twice pulled himself off the ground to save Volland's follow-up efforts.
Oblak's work for the night wasn't done as he still had to make saves from long-range efforts by Leon Bailey, Wendell and Kevin Kampl to prevent Atletico slumping to just a second Champions League home defeat under Simeone.
But the hosts held on to extend Spanish clubs' winning run over German sides in European competition to 11 ties.
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