after extra time; Inter squander 2-goal lead.
Atletico goalkeeper Jan Oblak was then the hero, saving penalties from Alexis Sanchez and Davy Klaassen, before Inter's Lautaro Martinez blasted his spot-kick over as the Spanish side advanced in thrilling style.
The draw for the quarter-finals takes place at 11:00 GMT on Friday, 15 March with Atletico's potential opponents being Arsenal, Barcelona, Bayern Munich, Manchester City, Paris St-Germain, Cheap Real Madrid Football Shirts and Borussia Dortmund, who beat PSV Eindhoven in Wednesday's other tie.
Two Champions League shootouts in two days
Inter, who lost 1-0 to Manchester City in last season's final, went first in the shootout but Oblak saved two of the first three kicks he faced. Atletico's Saul Niguez also missed, with Yann Sommer blocking his effort.
However, Memphis Depay, Rodrigo Riquelme and Angel Correa all converted their penalties to take Atletico into the last eight and complete a superb recovery.
Serie A leaders Inter won 1-0 in the first leg in Italy and looked to be cruising through when they doubled their overall advantage on Wednesday through Federico Dimarco's first-time finish following Nicolo Barella's pull-back on the left.
But Cheap Atletico Madrid Soccer Jersey pulled a goal back two minutes later, drawing level on the night.
Koke's pass was not cleared, Benjamin Pavard sliced his clearance and Antoine Griezmann reacted quickest to give his side hope of a memorable fightback.
Substitute Depay hit the inside of the post for Atletico as they looked to get the goal they needed.
It came soon after as Depay's low, angled shot in the 87th minute went in the bottom corner to the delight of the passionate home crowd.
Another Atletico substitute, Riquelme, had a glorious chance to win the tie but blazed over in the third minute of injury time as the match went into extra time, with neither side able to find the decisive goal before the drama of penalties.
Before this week, there had not been a penalty shootout in the Champions League since the 2016 final, when Real Madrid beat Atletico to lift the trophy.
Now there have been two in two days after Arsenal defeated Porto in similar fashion at Emirates Stadium on Tuesday.
Wasteful Inter lose for first time in 2024
Diego Simeone's Atletico are fourth in La Liga, 14 points behind leaders Real, and were eliminated by Replica Athletic Bilbao Football Shirts in the semi-finals of the Copa del Rey.
They have never won the European Cup or Champions League, losing in the final in 1974, 2014 and 2016, but kept their hopes alive this season as they became the first side to defeat Inter in 2024.
The Italians, led by coach Simone Inzaghi, have been in superb form and are 16 points clear at the top of Serie A as they look to become domestic champions for the 20th time.
On that basis, and with the lead they carved out, Inzaghi will know his side should have safely come through this contest.
Not only did they go out after holding a 2-0 advantage, but they twice had glorious opportunities to win the tie late on in 90 minutes.
Marcus Thuram and Barella failed to convert chances when faced with only Oblak to beat, and that proved costly as Atletico held their nerve in the shootout to book their place in Friday's draw.
