Dani Olmo had opened the scoring for Leipzig in the 51st minute and Joao Felix equalised for Atletico from the penalty spot in the 71st.
The result ended yet another title run for Atletico, who were seen as favourites to reach the semifinals but again will endure a disappointing elimination.
Diego Simeone's team knocked out defending champions Liverpool in the last 16 and were looking to return to the last four for the first time since 2017, a year after they lost their second final in three seasons.
Leipzig, founded 11 years ago with investment by Red Bull, will next face French powerhouse Paris Saint-Germain.
The Champions League is finishing the season amid the coronavirus pandemic with a last eight tournament in Lisbon. The semifinals will be next week and the final will take place on August 24.
Making only their second Champions League appearance, Leipzig looked more dangerous than Atletico during most of the match at the Jose Alvalade Stadium.
Neither team created many significant chances but Leipzig threatened the most with their quick passing and player movement.
Some of that fast action in front of the area led to Olmo's close-range header after a cross from the right.
Atletico Madrid improved after Portuguese forward Joao Felix came off the bench in the second half, and he equalised from the penalty spot after being fouled inside the area.
Atletico seemed in control after equalising but couldn't stop the late breakaway that ended their title hopes again.
"I'm convinced we gave everything we had," Simeone said. "But it wasn't enough. We have to raise our heads and try again next season."
Adams, who entered the match in the 72nd minute, picked up a pass from the left and sent a low right-footed shot that deflected off Atletico defender Stefan Savic and took keeper Jan Oblak out of action.
"It's a deflection but it counts as a regular one so I will take it," Adams said.
Adams joined Leipzig in January 2019 from another club owned by the Red Bull organisation, the New York Red Bulls of Major League Soccer.
Leipzig were making their debut in the knockout round after one appearance in the group stage of the elite European competition.
They eliminated last season's runners-up Tottenham in the round of 16 but entered the last eight without top scorer Timo Werner after he joined Chelsea.
"Tonight we're happy, tomorrow we'll start looking at [PSG]," Leipzig coach Julian Nagelsmann said. "That will be another difficult game but we'll have a plan."
- AP