The visitors defended their 1-0 lead from the home leg without any major scares and indeed went closest to scoring themselves when Nemanja Gudelj's first-half free-kick hit a post.
Anzhi, bottom of the Russian Premier League, finished the match with 10 men following the injury-time dismissal of Karlen Mkrtchyan.
The Russians started with the greater purpose, Gia Grigalava firing over and Oleksandr Aliyev shooting wide as they looked to wipe out their first-leg deficit early on.
The Dutch outfit struggled to get going and Fedor Smolov volleyed over Esteban Alvarado's goal.
AZ came closest to breaking the deadlock in the 33rd minute, though, as Gudelj's superb free-kick from left of centre struck a post.
The visitors threatened again soon after the break as Aron Johannsson, whose penalty decided the first leg, unleashed a 20-yard drive which was tipped over by Mihail Kerzhakov.
And the Anzhi goalkeeper came to his side's rescue again on the hour mark, producing a fine reflex parry to keep out the same player's shot from inside the area as the hosts had to start taking more risks.
At the other end Serder Serderov had a shot blocked by Etienne Reijnen, but the home side's frustration got the better of them two minutes into stoppage time.
Mkrtchyan saw a straight red for a foul on Steven Berghuis as Anzhi's European campaign came to an end.