Rafael Leao broke the deadlock for Milan just before half time with an elegant first-time finish from Theo Hernandez's pass, but in a swift response, an unmarked Koopmeiners received a ball from Emil Holm near the edge of the box before guiding a low shot into the far corner.
Koopmeiners then completed the comeback for Atalanta in the 59th minute when he converted a penalty after Alex Jimenez fouled Aleksei Miranchuk.
Atalanta manager Gian Piero Gasperini was sent off late in the first half after a heated conversation with the referee that persisted despite an initial warning.
The closing minutes intensified as a desperate Milan threw everything forward, but Atalanta keeper Marco Carnesecchi stopped attempts from Christian Pulisic and Yunus Musah.
In the dying seconds, Milan's hopes were raised as an alleged handball by Miranchuk inside the box underwent a VAR check but it was dismissed.
Milan's backup goalkeeper Antonio Mirante received a direct red card from the bench for dissent following his protest against the decision.
Atalanta will play Fiorentina in a two-legged semi-final in April following Vincenzo Italiano's side's win over Bologna on penalties on Tuesday.