After a poor first half, Dani Olmo headed the German side ahead but Atleti substitute Joao Felix levelled the tie from the penalty spot after he had been fouled in the box.
Madrid, chasing a third final in the last six years, seemed content to take the one-legged tie in Lisbon to extra time and sat back for much of the game.
They were punished for their characteristically defensive approach as Tyler Adams’ deflected strike two minutes from time sent Julian Nagelsmann’s exciting side through to face Paris St-Germain.
No Werner, no problem
Leipzig came into the tie without the focal point of their attack after Timo Werner’s £54m transfer to Chelsea earlier in the summer.
Werner scored 34 goals for the club this season while the starting XI in their first quarter-final at this level had mustered just 44 between them.
A lack of cutting edge was as evident as was expected, with Leipzig’s enterprising approach play all-too-often leading to nothing.
The German side were the only team playing progressive football in a dull first period – and that sense of adventure included centre-back Dayot Upamecano frequently carrying the ball out from a back three and suggesting he is a defensive prospect out of the very top drawer.
Their deserved lead arrived early in the second half when Marcel Sabitzer’s fine cross was met by Olmo’s late surge from midfield and he flicked a header beyond the dive of Atleti’s renowned keeper Jan Oblak.
If Leipzig’s lack of threat was down to the absence of an individual, their opponents’ issues were more systemic.
Former Chelsea striker Diego Costa was marooned up front and was bullied by young Frenchman Upamecano at times.
Just as in PSG’s fightback on Wednesday night, though, the introduction of a £100m+ substitute made the difference.
Portugal playmaker Felix was excellent as soon as he entered the match and won a penalty with a driving run into the area then confidently converted the penalty.
However any sense of a change in momentum was stymied by Atleti boss Diego Simeone, who appeared content to play for extra time as he instructed his side – unbeaten in 18 games – to sit in.
But a brilliant pass from Sabitzer freed Angelino into space and after he found Adams, the United States midfielder’s powerful shot flew in off the heels of Stefan Savic past a stranded Oblak.