Enda Stevens opened the scoring for the hosts.
Ibrahima Diallo met Oriol Romeu's cross after 23 minutes to make it 1-1.
Mohammed Salisu then put the Saints ahead before Oli McBurnie equalised for the hosts.
Despite Armando Broja having his penalty saved, the Premier League side were able to convert all their remaining penalties.
Romeu stepped up to score the winning penalty as Ralph Hasenhuttl's side prevailed 4-2 in the shootout.
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