Fatih Tekke started as striker but failed to make an impact.
His only serious attempt on goal was a long-range shot in the 60th minute - just before he was replaced by Pogrebnyak.
Instead, it was Arshavin who looks most likely to threaten Rangers.
The playmaker started on the left wing but regularly drifted infield and was at the heart of most of Zenit's attacks, despite playing with his right knee taped.
AZ went ahead when Ari broke free in the 26th and fed Graziano Pelle, who slid the ball into the far corner past goalkeeper Vyacheslav Malafeev.
Six minutes later, AZ exposed a possible Zenit frailty at set pieces when Hector Moreno climbed above the defense to easily head home Simon Poulsen's corner to make it 2-0.
On the stroke of half-time, Zenit pulled back a goal thanks to a quick exchange of passes inside AZ's penalty area.
Dominguez crossed from the right and Arshavin headed back to Konstantin Zyryanov. He slipped the ball to Roman Shirolov, who curled a low shot around Boy Waterman.
Arshavin had a string of chances after the break but saw substitute goalkeeper Joey Didulica save two shots and watched as his cross beat Didulica but bounced off the top of the bar.
That set the stage for Pogrebnyak to calmly slot in his penalty after being brought down in the area by Ryan Donk.