A first-half penalty from Aron Johansson settled the contest in AZ's favour and helped them erase memories of a 5-0 thrashing by the Russians when the two sides met in Alkmaar in last season's Europa League play-offs.
A cagey opening at the AFAS Stadion was not made any easier for Dick Advocaat's hosts by an injury to Steven Berghuis, who had to be replaced in midfield by Johann Berg Gudmundsson after 23 minutes.
And Anzhi immediately tried to exploit the situation, with Andrei Yeschenko's low cross into the area poked just wide of the target by Alexander Bukharov.
However, the Russians then gave away a penalty on the half hour when an inswinging corner from Gudmundsson struck the hand of Ali Gadzhibekov.
Johansson took the spot-kick and planted the ball low into the centre of the net, despite the best efforts of Mihail Kerzhakov to save it.
It was the goal the game so badly needed and Anzhi responded with a header from Bukharov, which fell past the left post, before a free-kick from Oleksandr Aliyev six minutes before the interval rebounded off the crossbar.
Bukharov threatened again moments later, forcing AZ goalkeeper Esteban Alvarado to block his 12-yard strike.
Kerzhakov then came to the rescue at the other end, palming away a drive from distance from Nemanja Gudelj which was on course for the top right corner.
Anzhi cranked up the pressure at the start of the second half and Esteban had to be alert to keep out a close-range header from Nikita Burmistrov.
Further chances came and went for Bukharov and Aliyev, while Celso Ortiz failed to inspire AZ with an under-hit effort after 69 minutes which trickled past the left post.
Johansson also had the opportunity to add to his tally nine minutes later but fired a shot straight into the hands of Kerzhakov from the edge of the area.
Roy Beerens soon followed up with a strike from distance for the Dutch hosts which cleared the bar, but the final minutes failed to produce any fireworks, leaving AZ with the narrowest of leads to defend in Russia next week.