The top-seeded Russians now have 10 points from five matches while Armenia surprisingly moved into second place with eight points, ahead of both Ireland and Slovakia who play their qualifiers later on Saturday.
Russia have now failed to score for a third consecutive match after losing two friendlies to Belgium (2-0) and Iran (1-0) in the last five months.
After a slow opening, the Russians took control, creating several scoring chances but failed to find the target.
Yura Movsisyan wasted Armenia's best chance early in the second half when he fired wide from close range.
Alexander Kerzhakov, who scored for Russia in each of their two previous qualifiers against Ireland and Macedonia, missed the top corner when he was sent clear a few minutes later.
Russia captain Andrei Arshavin had a quiet game, missing the target with his only shot at goal late in the game.
Coach Dick Advocaat said he was happy enough with a draw.
"Both teams were looking for a goal but we had a lot more chances than them," the Dutchman told a news conference.
"Unfortunately, we were missing the final pass and couldn't get a good shot on goal. Still, this draw is a good result, especially as we were playing away from home."
The two teams meet again in their next qualifier on June 4.
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