Holders Ajax reach Dutch Cup semis
Fouad Idabdelhay put Breda ahead after eight minutes when he broke through the Amsterdam defence and beat goalkeeper Maarten Stekelenburg.
Siem de Jong equalised with a long drive 10 minutes later and was fouled before the break, giving Serbia midfielder Sulejmani the opportunity to put the home side ahead from the penalty spot.
Ajax dominated the rest of the match but had to wait until the 82nd minute for their third goal when Vurnon Anita netted a fluent combination.
Sulejmani added his second a minute later.
Twente Enschede beat PSV Eindhoven 7-6 on penalties on Wednesday after regular time ended 1-1 and were joined by Utrecht in the semi-finals after they beat Groningen 3-2.
Second division RKC Waalwijk brushed aside the amateurs of Noordwijk with a 6-0 home win on Tuesday.