Stuttgart took an early 2-0 lead through Eliot Bujupi and Efe Korkut, but Liverpool fought back with goals from Kieran Morrison and Trent Kone-Doherty.
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Despite chances for both sides, the match ended 2-2, leading to penalties.
Amara Nallo missed the decisive spot-kick, sending Stuttgart through.
The game featured early goalkeeping heroics from Kornel Misciur and end-to-end action throughout.