The visitors went down 3-1 in the fourth-round clash as two goals from Guus Hupperts sent Roda through.
Hupperts put the hosts in front in the 66th minute only for Patrick van Aanholt to equalise.
But substitute Marc Hocher restored Roda's lead in the 83rd minute, just one minute after coming on, and Hupperts sealed the win with his second in the 90th minute.
Feyenoord, with coach Ronald Koeman absent, edged past Heracles Almelo on penalties after a 1-1 draw.
Koeman was granted compassionate leave after his father was hospitalised following a heart attack, meaning assistants Jean-Paul van Gastel, Giovanni van Bronckhorst and Patrick Lodewijks took charge for the visit to Almelo.
Graziano Pelle put the Rotterdammers ahead in the 12th minute, but Bryan Linssen levelled in the 27th minute.
Feyenoord had Stefan De Vrij sent off four minutes into extra-time, but they held on before claiming a 6-5 win on penalties.
AZ Alkmaar also went through on spot-kicks, seeing off Heerenveen after a thrilling 2-2 draw.
Hakim Ziyech put Heerenveen ahead in the ninth minute, but Aron Johannsson levelled and then put AZ ahead in the 105th minute.
Ziyech scored a 120th-minute penalty to force a shootout, but the hosts triumphed 7-6 on penalties to go through.
Elsewhere, FC Utrecht won 2-0 at Cambuur Leeuwarden thanks to goals from Jens Toornstra (13) and Steve De Ridder (79), while JVC Cuyk beat MVV 2-1.