The Eredivisie outfit were outplayed for large spells of the match at the AFAS Stadion but claimed victory thanks to Aron Johannsson's first-half penalty.
Ali Gadzhibekov conceded the spot-kick after handling in the area and it was fitting that AZ's conversion also needed a slice of luck.
Anzhi goalkeeper Mikhail Kerzhakov got both hands to Johannsson's effort but saw the shot ricochet off his arm and in.
The Russian Premier League outfit sit bottom of the table and only produced their first win of the league season on Sunday.
But they produced an excellent performance that defied their league standing, with Olexandr Aliyev going closest when he hit the bar from a fierce 30-yard free-kick, while AZ goalkeeper Esteban was on top form.
And Anzhi, who beat AZ 6-0 on aggregate in the 2012-13 Europa League play-offs, will take confidence from the display ahead of the second leg, to be played in Russia next week.
After a tame beginning, the game burst into life in the 15th minute as Fedor Smolov was controversially denied for Anzhi.
Smolov quickly pounced after strike partner Aleksandr Bukharov had a shot blocked, but his looping header was clawed away by Esteban, who could only palm the ball upwards before scrambling the rebound away.
Anzhi players thought Smolov's effort had crossed the line but their appeals were turned down, before Bukharov stabbed wide from Andrei Eshchenko's cut-back.
AZ took the lead against the run of play, though, after Gadzhibekov handled Jeffrey Gouweleeuw's header in the penalty area with 28 minutes gone.
Johannsson, who has 16 Eredivisie goals this season, converted from the spot but Anzhi kept up the pressure, as Aliyev struck the woodwork with an incredible free-kick.
Aliyev's effort beat Esteban but thudded into the crossbar, and Bukharov then forced a great one-on-one save from the goalkeeper after being played through by Smolov.
Nemanja Gudelj stung the palms of Kerzhakov with a long-range effort before the break but the hosts could count themselves lucky to lead at half-time.
Aliyev went close with two efforts just after the interval, first shooting wide from 18 yards before he fired over from distance, and Nikita Burmistrov was next to test Esteban, with his back-post header from Aliyev's cross well saved.
AZ manager Dick Advocaat was sent to the stands as he grew anxious but Anzhi tired as the match grew older, and Johannsson tested Kerzhakov with a late effort.
One goal was enough for AZ, though, who have now won three matches in a row without conceding.
