Russia have the threat of a six-point deduction hanging over them in next European Championship qualifying if their fans cause trouble of the same severity that marred the crowd disturbances in their 4-1 victory over the Czech Republic in Wroclaw.
"Maybe they should be more calm," Dzagoev told reporters on the eve of Russia's final Group A game against Greece in the Polish capital on Saturday.
"We don't need to lose six points in qualifying and we don't want anything bad for our national team."
Russian and Polish fans also clashed before the second Group A game in Warsaw on Tuesday, prompting football's European governing body UEFA to open another disciplinary case.
Prickly Russian coach Dick Advocaat berated a Dutch reporter for asking him to comment on the fighting that took place outside the National Stadium, and his mood appeared not to lift for the remainder of the news conference.
The Dutchman claimed after the 1-1 draw with the Poles that his side had played the best football at Euro 2012, but picked holes in that performance on Friday.
"In the second half we did not play that great against Poland, but that had to do with losing the ball unnecessarily," he said.
"The first half was very good. There were not too many chances for either team and in the second half Poland did a lot better. We simply have to get better."
Dzagoev, who scored twice against the Czechs and headed Russia in front against Poland, owned up to being culpable of wasting possession.
"I don't think that I have played two good games. Yes, I scored against Poland but I lost possession a lot as well, said Dzagoev, who celebrates his 22nd birthday on Sunday.
Russia top Group A with four points and need just a draw against Greece to reach the last eight.
Advocaat said midfielder Konstantin Zyryanov had been sick on Friday but otherwise had a "fit and healthy squad".
The coach backed misfiring striker Alexander Kerzhakov who has missed a glut of chances in Russia's two games.
"Kerzhakov is a player who is going to take his opportunities and he can score a goal out of nothing, even though he may not play that well," Advocaat said.
"He can score goals and has done that his whole career."
