Barry van Galen was sent off in the 24th minute for elbowing Alexander Ostlund in the penalty box.
He was followed four minutes later by Kenneth Perez who handled on the line as Feyenoord seemed certain to score.
Henk Timmer saved Dirk Kuyt's penalty.
However, it was Kuyt who scored the opener four minutes later as he latched onto a cross from Basto.
Lazovic scored Feyenoord's second four minutes before half-time - he nipped in on a rebound from a Kuyt shot.
In the second half Alkmaar had the better of the game - with more and better chances as Feyenoord struggled to make the most of their numerical superiority.
AZ left Stein Huysegems upfront but supported him well when they could with De Cler, Landzaat and Meerdink pushing up from midfield. Impressive defender Jan Kromkamp made several impressive one-man charges up the field which had Feyenoord in tatters.
With 20 minutes to go Co Adriannse threw on three attacking subs and it nearly paid off as skipper Denny Landzaat scored with 7 minutes to go.
It was all in vain however as Kuyt scored a third for the visitors with a header following a short corner.