Toivonen hit a brace in last week’s opening 3-1 win against Heerenveen.
De Graafschap played 85 minutes with 10 players after Purrel Frankel had been red-carded for a fifth-minute foul which allowed Balazs Dzsudzsak to open the scoring from the penalty spot.
Toivonen struck in the 25th, 47th and 55th minutes before Nordin Amrabat and Ibrahim Afellay completed the rout.
Martin Jol’s Ajax warmed up for Tuesday’s Champions League play-off trip to Dinamo Kiev with a 4-2 win over Vitesse Arnhem.
Davy Propper gave Arnhem the lead before Jan Vertonghen and Gregory van der Wiel put Ajax 2-1 up.
Marco van Ginkel levelled for the visitors before Ajax made sure of their first win of the season with goals from Siem De Jong and Vurnon Anita.