Anzhi, playing without captain Samuel Eto'o, scored first-half goals through Bosnian defender Emir Spahic and Ivory Coast striker Lacina Traore before Brazilian playmaker Willian added the third late in the second half to complete the rout.
The only worry for coach Guus Hiddink was an injury to Traore which forced the tall striker to leave the pitch late in the game.
"He pulled a hamstring and usually it takes several weeks to heal such an injury," Hiddink told reporters.
"We've had a bit of a slump recently so we were really motivated coming here (to Nizhny Novgorod)," the Dutchman added.
"We decided to rest Samuel Eto'o because he was still carrying a slight injury coming back from the (Cameroon) national team. He played last week but today we decided to give him some rest so he would be fully fit for our next matches."
The win, their first since the league's three-month mid-season break which started in early December, put Anzhi within 10 points of leaders CSKA Moscow, who were held to a 0-0 draw by city rivals Dynamo on Friday, with six games remaining.
Champions Zenit St Petersburg are second, six points off the pace, after beating Lokomotiv Moscow 1-0 on Saturday.