The violent, offensive and unsafe behaviour of Russian supporters has already landed the Football Union of Russia a huge fine and a suspended six-point deduction for their Euro 2016 qualifying campaign.
Any repeat on Saturday could see that deduction enacted by UEFA and CSKA Moscow midfielder Dzagoev said of fans: "Maybe they should be more calm.
"We don't need to lose six points in the next qualification so we don't want anything bad for our national team."
Manager Dick Advocaat reacted angrily to questions about Russia fans' conduct in Tuesday's 1-1 draw against Poland.
"Are you really going to ask me these types of questions?" he said.
"Tomorrow [Saturday], we're going to be playing a very important game and you're going to ask me questions like this?"
A draw on Saturday would be enough for Russia to qualify for the quarter-finals, while even a narrow defeat could be sufficient depending on the outcome of the Czech Republic-Poland game.
Dzagoev said: "I'm really sure that we will go through and we are very focused. We must go on the pitch and win the game.
"We don't feel relaxed. We simply must win."
Advocaat added: "Whenever you go and play a game, you have to win and you have to want to win.
"We are very confident in ourselves. I'm sure the Greece coach will say the same thing."
Dzagoev scored against Poland and another good display could put him in the shop window.
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The 21-year-old said: "Of course, I am willing to try myself in other clubs in Europe.
"But I will think about it only after the tournament."
Advocaat confirmed forward Aleksandr Kokorin would miss the rest of Euro 2012 after suffering a torn hamstring in training in Sulejowek on Friday.
Konstantin Zyryanov also missed Friday's training session through sickness and was a doubt for Saturday.
Greece must win to stand any chance of progressing and head coach Fernando Santos said: "My faith in my team is total.
"It's the same faith I had back in the autumn after our (qualifying) draw in Latvia.
"Back then, it was tense and we needed a victory against Croatia to have any chance of qualifying.
"I believed in that team, I said so, and afterwards I was vindicated. I am saying the same now."
Santos urged his players to forget Tuesday's Czech Republic defeat.
"We must leave it all behind us," he said.
"We must take all our passion on to the pitch and I am sure we will make the Greek fans happy."
Greece have conceded early on in both games so far and Santos added: "It is something I find hard to understand.
"Our preparation has been spot on and the players are motivated.
"The only thing I can do is point out the mistakes to the players, but they are perfectly aware. I am sure that they will show more intensity in the next match and everything will go fine.
"I am considering some changes to the line-up as I am looking for quick thinking when in possession and pressure when we're not."