But Antoine Griezmann immediately pulled the hosts level on the night and three minutes before the end of the regulation 90 minutes substitute Memphis Depay took the tie to extra time at 2-2 with a fine low finish.
That set up the decisive shoot-out which ended with Inter missing three of their five penalties, star striker Lautaro Martinez handing passage to the last eight to Atletico when he booted his spot-kick way off target.
“I think penalties are a lottery. Today luckily I stopped two,” said Atletico goalkeeper Jan Oblak, who saved from Alexis Sanchez and Davy Klaassen.
“It’s not easy to stop a penalty. Sometimes it looks easy from the outside but it’s not.”
“I’m very, very happy for the team because I think we did a great match,” he added. “We played well. Maybe in the last month we’ve not been in great shape but we showed we can do much more.”
The two teams could barely have come into Wednesday’s match in more contrasting form as Inter had won 13 matches on the bounce since the start of 2024 while Atletico had won only one of their last five.
However it was an even and exciting contest in the Spanish capital in which Atletico pushed hard for the win and both sides created a clutch of opportunities.
Home advantage turned out to be crucial for Atletico who have won 19 of their 21 fixtures at the Metropolitano in all competitions.
Atletico are also unbeaten in 17 knockout matches on home turf in the Champions League and in front of a passionate crowd they fought back from an impossible looking position against runaway Serie A leaders Inter.
Atletico fight back
Depay smashed a powerful effort off the woodwork moments before his strike took the two teams to an additional 30 minutes while Rodrigo Riquelme wasted a glorious chance to snatch the tie for Atletico without the need for extra time.
Inter meanwhile were left ruing two great chances on the break as Atletico pushed for their second, both Marcus Thuram and Nicolo Barella finishing poorly with chances to seal the tie.
Simone Inzaghi’s Inter now turn their attentions back to winning their 20th Serie A title while Atletico await Friday’s draw for the last eight in Nyon.
“Once we went ahead we needed to be better, those three-four minutes in order to maintain that advantage Dimarco gave us,” said Inzaghi.
“At that moment the match seemed in our hands but instead they got back into the game.”
Inter, who wasted a series of opportunities to earn a bigger first-leg advantage, are 16 points clear of AC Milan with 10 league matches remaining.
It was a fast-paced encounter from the off as Atletico pushed immediately to level the tie with aggressive pressing.
Samuel Lino burst down the left flank and tested Yann Sommer in the fifth minute before his counterpart Denzel Dumfries forced a double save from Oblak.
Inter were dangerous on the break and it was with a quick-fire exchange of passes that the away side took the lead, Dimarco confidently netting first time from Barella’s neat cut-back.
However Inter’s celebrations were almost instantly cut short when Griezmann pounced after Koke’s through ball deflected off two Inter defenders, taking his club-record goal tally to 176.
Griezmann forced a decent stop from Sommer shortly after the break as Atleti again tried to pen Inter into their own area.
Thuram blasted over after being perfectly put through by Martinez, leaving Inzaghi raging on the sidelines, before Barella shot weakly at Oblak in similarly good position.
Depay then struck and all the momentum that had been building went definitively in Atletico’s direction, eventually leading to a joyous shoot-out win.
© Agence France-Presse
Photo: Twitter @CBSSportsGolazo