At the end of a balanced first half, in which Ron Vlaar hit the post for Feyenoord with a long drive, De Zeeuw opened the scoring and seconds later Feyenoord midfielder Stefan Babovic was sent off for elbowing Jan Vertonghen.
Urby Emanuelson doubled the lead two minutes after the break with a fierce volley and Gregory van der Wiel added a third before Denny Landzaat pulled one back with a superb lob.
De Zeeuw's second restored the three-goal lead, while in the final minute Feyenoord goalkeeper Rob van Dijk was sent off for a professional foul on Luis Suarez, who converted the penalty for his 16th league goal of the season.
Twente Enschede, 2-1 winners at Roda JC Kerkrade on Saturday, are top with 32 points from 12 matches, two above PSV Eindhoven who beat Vitesse Arnhem 1-0.
Ajax remain third with 29 and travel to leaders Twente next Sunday. Feyenoord are fourth with 23 points.
Utrecht wasted the chance to leapfrog Feyenoord as they suffered a 3-1 defeat at Heracles Almelo, while Sparta Rotterdam were 2-0 winners at home to NEC Nijmegen.
‘The pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
