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.
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.
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.
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.
