The dramatic late win marks the first time the Bundesliga club has reached the last four of the competition and sets up a clash with Paris Saint-Germain on Tuesday for a spot in the final.
With two minutes of normal time remaining, Adams snatched victory with a deflected shot from outside the box to set up the semifinal clash with the Parisians.
"It's all a bit surreal," Adams said after the match. "And now obviously facing such a world class team like PSG, we're excited to show what we're capable of. We're confident in our abilities, but we need to go out there and continue to show what we are capable of.
"I think with such a young team we're not really fazed in these moments, we just go out there and execute the game plan. What we were able to do tonight with different players in different positions on the field, it was really special."
The German side had taken the lead in the 51st minute with a header from Spanish forward Dani Olmo but a disjointed Atletico were shaken into life when record signing Joao Felix came off the bench, earning and scoring a penalty to level in the 71st.
Atletico, runners-up in 2014 and 2016, looked more likely to score the next goal but it was the club founded only eleven years ago who were left celebrating in front of the empty seats of the Estadio Jose Alvalade.
Now the team, built up from the lower leagues and owned by energy drinks company Red Bull, will come up against Neymar and company with a chance at a place in the Aug. 23 final.
As with all games in this 'final eight' mini-tournament, there were no fans to witness the drama but it was clear from the outset that Leipzig felt at home among European football's elite.
They fired an early warning shot when Marcel Halstenberg was picked out at the back post but his difficult volley flew off target.
Julian Nagelsmann's side quickly settled though into the brand of attractive passing and movement that has earned them many admirers.
Kevin Kampl was busy in the centre of midfield and left wing back Angelino was enjoying plenty of space down the flanks but it was the performance of French central defender Dayot Upamecano that was the most impressive.
Solid defensively, taking care of Atletico's over-isolated striker Diego Costa, he also had the confidence to bring the ball out of the back and be the starting point for Leipzig's moves.
But six minutes after the interval, they found a way through -- Marcel Sabitzer's cross from the right was met by Olmo, the Spaniard's angled header flashing past Jan Oblak and into the far, bottom corner.
The goal prompted Atletico boss Diego Simeone into a change, with Felix, the 20-year-old Portugal international and club record signing, introduced in the 58th minute.
The move paid off when Felix glided into the box and was brought down by an ill-timed tackle from Lukas Klostermann for a penalty.
Felix hasn't had the most impactful of debut seasons but he had the confidence to get to his feet and bury the spot kick to get the Spaniards back on level terms with 19 minutes left.
Atletico sensed the momentum had shifted and they pushed forward in search of a winner, which looked the more likely outcome, possibly in extra-time, as Leipzig struggled to regain control of the game.
But with two minutes of normal time remaining, substitute Adams tried his luck with shot from outside the box, which deflected off defender Stefan Savic and beat the wrong-footed Oblak.
"It's a deflection but it counts as a regular one so I'll take it," said Adams. "That's why it's so exciting for me to come in and play in such a big game in the Champions League. You know for the first six months when we started the Champions League campaign I was hurt, so it's a good feeling to get out there, be fully fit and showing them what I'm capable of.
"We're really, really excited. We know that Paris is obviously a great team; we saw what they did yesterday against Atalanta, they have strong players everywhere on the field, so we're going to need to match that energy.
"Now it's about recovering in the right way, getting our mentality right and preparing a game plan. We're confident in our abilities, like you saw today, so we're going to bring what we're capable of."
Atletico piled on the pressure, but Leipzig had the belief and the fortitude to hold on for their most famous victory.
Simeone after the match said the loss ends a hectic run for his side.
"We couldn't play the game as we wanted. I'm convinced we gave absolutely everything we had. It's been a really long year, a really tough year, stopping for 40 or 60 days, coming back to play eleven games in a row with the pressure of having to get into the Champions League no matter what, a week off, we came back to train, we tried to give everything, it didn't turn out, we have to lift our heads and get ready for next year," he said.
Information from Reuters was used in this report.