The German side had come from behind to attract 1-1 at home within the first leg last week and they went in front in the 18th minute in Bergamo as Konrad Laimer cut the ball back from the right for Nkunku to score.
Atalanta desired a penalty early in the 2nd 1/2 when a Ruslan Malinovskyi unfastened-kick seemed to strike an arm inside the wall in the area. However, Spanish referee Mateu Lahoz upheld his choice no longer to award a niche-kick after coming across to check the snap shots on the pitchside reveal.
An Atalanta equaliser could have forced extra time but Leipzig wrapped up the victory with three minutes left as Nkunku transformed a penalty he himself had won. That goal was the thirtieth of the season in all competitions for Nkunku, who made his complete international debut for France closing month.
Leipzig, who reached the Champions League semi-finals two years in the past, improve to a ultimate-four tie towards either Braga or Rangers. These aspects meet in Glasgow later on Thursday with Braga main 1-zero from the first leg.
Elsewhere on Thursday, Barcelona hosted Eintracht Frankfurt and Lyon confronted West Ham United with both ties poised at 1-1 following the primary legs.
