It started well for the visitors, with Kuyt opening the scoring just nine minutes in, from the penalty spot.
AZ were not behind for long, with Barry van Galen bringing them level just five minutes later before Ali Elkhattabi put them ahead just before the break.
Seconds after, Shinji Ono (pictured) was brought on for Danko Lazovic and the home side then went a man down when Peter Wijker was shown a second yellow card.
Despite a valiant attempt to cling onto their slender lead, Kuyt was on hand to score his second goal of the game and earn a point with 10 minutes remaining.
NEC Nijmegen emerged with all three points from their fiery home clash against Twente following a 2-1 victory.
Two players were dismissed in the clash as NEC overturned an early 1-0 deficit at the De Goffert.
Collins John got the visitors off to a flyer when he fired them ahead, and it got better shortly after when Dejan Govedarica was dismissed.
Twente were then down to 10-men six minutes later when Blaise N'Kufo followed Govedarica into the changing rooms for an early shower.
Within moments of the restart goals from Fabian De Freitas and Dennis de Nooijer turned the match around giving NEC the victory.
Elsewhere, NAC Breda stole a last-gasp point at home to Willem II thanks to Stefano Seedorf's goal.
The home side trailed early on, thanks to Denny Landzaat's seventh minute effort, before Seedorf, Clarence's nephew, made it 1-1 snatching an unlikely draw.
Click below for Dutch match summaries:
AZ 2-2 Feyenoord
NAC 1-1 Willem II
NEC 2-1 FC Twente
Click here for the latest Dutch Eredivisie table.
