The entertaining clash had finished 2-2 after extra-time but that was nothing compared to Vitesse Arnhem's 10-1 hammering of third-tier amateurs ADO '20 Heemskerk.
Graziano Pelle put Feyenoord ahead after 28 minutes with a well-executed volley but 10 minutes after the break Jeffrey Gouweleeuw equalised.
Midway through the second half, Lex Immers restored the lead with a close-range header followed by several missed chances for Pelle.
Heerenveen then took the initiative in the final 10 minutes of normal time and Filip Djuricic netted a rebound in stoppage time after a free kick from Lukas Marecek bounced back off the bar.
"It is always a pleasure to be decisive in a shootout but we caused our own problems," said Feyenoord keeper Erwin Mulder, who saved four penalties.
"After our second goal we should have sealed the match and then we again concede a goal from a set piece," he added.
Last season's runners-up Heracles Almelo had to come from behind to beat NAC Breda 3-1.
Arnhem thrashed Heemskerk thanks to doubles from Wilfried Bony, Nicky Hofs and Renato Ibarra.
On Tuesday, holders PSV Eindhoven cruised to a 4-0 away win against amateurs Rijnsburgse Boys.
Ajax will be in action on Thursday at Groningen.