Ted van de Pavert put Graafschap ahead after 15 minutes notching in a corner kick at the near post. Sulejmani equalised six minutes later.
Derk Boerrigter gave Ajax the lead after some fluent one touch football after 31 minutes and two minutes later team mate Siem de Jong had a penalty saved by Joost Terol.
Sulejmani and Theo Janssen settled the match just after the hour mark with two goals in three minutes.
A superb volley by Maarten Martens earned Alkmaar the lead against PSV before a 30 metre drive from Nick Viergever doubled the lead.
On the stroke of halftime Dries Mertens pulled one back but 10 minutes before time U.S. international Jozy ALtidore sealed Alkmaar's win, despite the sending off of Niklas Moisander.
On Friday Feyenoord opened the season with a 2-0 win at neighbours Excelsior Rotterdam.