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Leipzig, who were only founded in 2009, reached the semi-finals for the first time with a 2-1 win over Atletico, twice runners-up.
They will face Paris Saint-Germain for a place in the final.
Holding midfielder Adams scored the winner in the 88th minute, his first goal for the club, after a Joao Felix penalty had cancelled out Dani Olmo's opener for Leipzig.
"I am not your typical goal scorer, but I am happy I could help the team," said Adams, who came on as a substitute.
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Adams' effort came after his fired shot took a cruel deflection of Stefan Savic and flew in.
"Coming into this game, I was told I wasn't starting but needed to be ready to come off the bench and make an impact, so I had to get myself in the right mindset to come out and help in anyway possible," added the 21-year-old.
"We are really excited, we know Paris Saint-Germain are a good team and we saw that yesterday what they did against Atalanta, with the players they had on the field, so we need to match that energy.
"Now it's about recovering in the right way, getting our mentality and preparing a game plan, but we are confident in our abilities like you saw today and we will bring what we are capable of."
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