The hosts enjoyed the bulk of possession but chances were few and far between as Russia, who beat Ireland 3-2 in Dublin last October, stayed top of the standings with 17 points from eight matches - two ahead of the Irish.
With two matches left Russia's fate is still in their own hands although Slovakia, who Russia face next month, can draw level on points at the top if they beat Armenia on Tuesday.
Russia coach Dick Advocaat was left to rue missed chances.
"I can't be happy with tonight's result," the Dutchman told a news conference. "We controlled the game for 90 minutes and had seven or eight clear chances but still couldn't score at least once. But we still control our own destiny."
Igor Semshov, who scored Russia's lone goal in a 1-0 win over Macedonia last Friday, missed the best opportunity of the first half when his low drive from close range had Irish keeper Shay Given beaten but was cleared off the line by Richard Dunne.
The Russians increased the pressure after the break, forcing the visitors on the defensive with quick runs down the wings on the slick artificial pitch at the Luzhniki stadium.
Semshov wasted another chance after the restart while substitute Diniyar Bilyaletdinov fired high over the bar from a good position shortly after coming on late in the second half.
Given rescued his team late in the game when he blocked a diving header by Konstantin Zyryanov with his feet as the Irish hung on for a point that keeps them in contention.
Giovanni Trapattoni's side, missing John O'Shea, Kevin Kilbane and Shane Long through injury and Sean St Ledger through suspension, rarely troubled Russia's reserve keeper Vyacheslav Malafeyev, who was playing in place of injured Igor Akinfeyev.
Trapattoni remained optimistic about his team's chances, especially as their last two games are against Andorra and Armenia. "We still have two more games," said the Italian coach.
"Russia are in first place for now but they must travel to Slovakia so anything is possible."
