It was a disappointing night for Diego Simeone's side who had boasted a 26-match unbeaten run in the league, stretching back to February.
Atletico, which has already beaten Barcelona this season, visits Real Madrid next weekend.
A Diego Maradona jersey was put alongside the Real Madrid reserve players, and a moment of silence was observed before the match as a tribute to the Argentina great who died this week.
Diego Costa asked to be substituted for what seemed to be a left thigh injury early in the second half.
Simeone, 50, is isolated at home and has not presented symptoms.
Leipzig, founded 11 years ago with investment by Red Bull, will next face French powerhouse Paris Saint-Germain, which eliminated Atalanta in Wednesday’s quarterfinal in Lisbon.
We played well against great team: Atletico Madrid manager Diego Simeone after draw against Barcelona
Atletico Madrid holds the third spot on the La Liga table with 59 points, only behind Barcelona and Real Madrid.
If Liverpool remain favourites to go through, the roar from the Atleti fans at the final whistle, aimed at their fist-pumping players on the pitch, suggested Simeone's team now have belief.
Diego Simeone's side could be overtaken on Sunday by Sevilla, who visit Celta Vigo, but the coach welcomed the return to form.
The club is currently placed on the sixth position on La Liga table and Simeone stressed that they want to go ahead game by game.
The team that finished the last two league seasons as runner-up to Barcelona and has its fans accustomed to competing for titles is now in danger of having nothing to play for.
One victory from their opening four league games left Atletico seven points adrift of Barcelona a fortnight ago but three on the spin have now steered Diego Simeone's side back on track.
Diego Simeone has claimed if Lionel Messi played for Atletico Madrid, his side would have beaten Barcelona on Sunday.
Griezmann turned his back on interest from Manchester United to sign a new contract after Atletico lost an appeal against a transfer ban.
Atletico Madrid has fallen to Real Madrid in the quarters, semis and twice in the final in the past four seasons.