Federico Dimarco opened the scoring in the 33rd minute to double Inter's aggregate advantage, before Antoine Griezmann quickly halved the deficit on his return from injury.
Inter looked set to hold onto their slender lead, but they could not stop Memphis Depay from netting a late leveller and sending the tie to extra time.
With neither team able to find a winner in extra time, a Champions League last-16 tie was settled by penalties for a second night in a row.
Atletico dispatched three of their four spot-kicks, while last season's runners-up could only convert twice from five penalties, ensuring the hosts sealed their place in the last eight for the second time in three seasons.
Inter travelled to the Spanish capital off the back of 13 consecutive victories, including a narrow 1-0 win in February's first leg against Atleti.
Despite enjoying a sensational run of form, the Nerazzurri knew they would have their work cut out to get the better of an Atletico side that had lost just one of their previous 31 competitive home games prior to the second leg.
Samuel Lino threatened to open the scoring inside the opening five minutes, unleashing a left-foot strike to force Yann Sommer to make a save low down to his left.
Sommer's opposite number Jan Oblak was then called into action to make two strong saves in quick succession to keep Denzel Dumfries at bay.
Not long after, the Atltico goalkeeper was required to quickly back-peddle and stop Lautaro Martinez from finding the net with the help of a wicked deflection off Alex Witsel.
In a gripping opening half-hour, it was then the turn for Alvaro Morata to go close, as he rose high to meet Mario Hermoso's lofted cross, only to see Sommer comfortably gather his headed effort.
Ultimately, it was the visitors who found the breakthrough in the 33rd minute, with Dimarco slotting home from Nicolo Barella's neat pass to round off an incisive move.
However, Inter's two-goal aggregate lead lasted just two minutes before Griezmann netted his 18th goal of the season.
Benjamin Pavard's sliced clearance ricocheted off Alessandro Bastoni and into the path of Griezmann, who quickly swivelled to send the ball beyond Sommer.
Griezmann was presented with the chance to pull Atleti level in the opening stages of the second period, but he could find no way past Sommer after being teed up by Marcos Llorente.
As the contest entered the final 15 minutes, the Serie A leaders spurned a couple of chances to put the tie beyond doubt.
Marcus Thuram blazed over from a promising position, before Barella fired straight at Oblak following a swift counter-attack.
Inter were ultimately left to rue those missed chances when Depay restored parity just moments after thumping the inside of the post.
The Dutchman collected Koke's pass with his left foot and unleashed a driven shot with his right to register his first Champions League goal of the season.
With momentum in Atleti's favour, Rodrigo Riquelme looked destined to score a 93rd-minute winner, but he could only send his golden opportunity into the stand behind Inter's goal.
Depay went closest to finding a winner in stoppage time, with the Dutchman drawing a fine save from Sommer after latching onto Riquelme's low cross.
Sommer made another impressive stop in the subsequent penalty shootout, but it ultimately proved to be in vain, as Oblak made two crucial saves before Martinez blazed over the decisive spot-kick.
Atleti's penalty-shootout success ensures Spain will have three representatives in the quarter-finals, while Inter's exit means there will be no Italian side in the last eight.