Frank de Boer's men won both meetings between the clubs last season, helping them claim a fourth straight Eredivisie title.
But neither team has made a confident start to the new campaign and that contributed to a frantic opening in Rotterdam as the hosts twice struck the bar in the opening three minutes.
It was the visitors who took the lead, however, as Ricardo van Rhijn curled in a stunning free-kick from 25 yards after four minutes.
Feyenoord struck the crossbar for a third time 10 minutes later and created countless chances in the second half but found Jasper Cillessen in inspired form, with Ajax hanging on for a fifth consecutive win over their fierce rivals and one that could kick-start the champions' season.
AZ's season took an unexpected turn in midweek when head coach Marco van Basten stepped down citing "mental and physical" problems, having only taken charge at the start of the campaign.
His assistant Alex Pastoor turned down the chance to take on the role, with Dennis Haar at the helm for the 1-0 win over in-form PEC Zwolle.
Marcus Henriksen smashed in the only goal of the game from a 72nd-minute corner.
Cambuur started the game in third spot but their task became tougher when Lucas Bijker was sent off four minutes before half-time and Jeffrey Bruma and Adam Maher took advantage after the break.
Elsewhere, Utrecht beat Groningen 1-0 at home thanks to Ruud Boyman's 61st-minute goal. The visitors could have levelled eight minutes later but Tjaronn Chery missed his penalty.
