The Eredivisie outfit won the first leg 1-0 and a spirited showing in Russia was enough to send them through to the next round
Anzhi, who saw Karlen Mkrtchyan dismissed late on, could count themselves unfortunate to come into this encounter 1-0 down after dominating large spells of the first leg, but posed little threat as they exited the competition.
Nemanja Gudelj hit the post from a first-half free-kick for AZ, who committed few men forward in a bid to hold on to their lead.
The Russian Premier League's basement club exerted some pressure late on in a bid to salvage their European dreams, but the Dutch defence held firm to extend their unbeaten run in the competition to 10 games.
Defender Gia Grigalava, one of four changes from the first-leg loss, found himself in an advanced position early on, but the Georgian wasted the hosts' first sight of goal by lofting his effort harmlessly over the crossbar from the edge of the penalty area.
AZ boss Dick Advocaat – watching from the stands after his sending off in Alkmaar – had plenty of room to make himself comfortable as a sparsely populated Stadion Saturn contributed to what was a low-key opening to the encounter, with both sides struggling to put either goalkeeper under any pressure at all.
Fyodor Smolov ballooned another speculative strike after 26 minutes to send ball boys scurrying in the empty stand behind the goal, before Anzhi goalkeeper Mikhail Kerzhakov got a rare touch as he got down to save a limp effort from Aron Johannsson, whose penalty settled the first leg.
Gudelj almost dashed Anzhi's qualification hopes nine minutes later when he hit the left-hand post from a curling free-kick just outside the box – the closest either side came to breaking the deadlock in a turgid opening period.
AZ made the early running in the second half, with Johannsson forcing a superb diving save from Kerzhakov, turning on the edge of the box before curling an effort towards the top left-hand corner of the net, only to see it expertly tipped over.
Anzhi had Kerzhakov to thank again just after the hour mark as he leapt to his right to keep out a Steven Berghuis drive across goal, and the AZ man prodded wide soon after as a tie-settling strike continued to elude the Eredivisie outfit.
However, it ultimately proved unnecessary, as Olexandr Aliyev's free-kick that whistled narrowly wide of the post late on was the closest the Russians came to finding the net.
And their night was further soured when Mkrtchyan saw red for hacking down Berghuis, who was clean through on goal after Kerzhakov had ventured forward for a late corner.