The USA international converted a penalty in the opening 45 minutes to help his side to a vital win in Alkmaar
The referee awarded the Dutch side a spot kick after an unfortunate handball in the first half and Aron Johannsson fired home from 12 yards to decide the game. 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.
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 coach 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.