The dollar-denominated RTS index had fallen by 1.12 percent by 1130 GMT, recovering from a 3.55-percent slide earlier in the day.
Russia's battered ruble fell to around 82 on Tuesday morning before bouncing back to 79.55 against the greenback.
The ruble last week hit an all-time record low against the dollar, falling past the 82-ruble mark.
Russia -- whose energy-reliant economy has already been pushed into recession by low oil prices and Western sanctions over Ukraine -- has been rocked by the fresh fall in oil prices and officials have come under increasing pressure to react.
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Russia's central bank chief Elvia Nabiullina said last week that authorities have "all the means" necessary to ward off instability.
President Vladimir Putin has said that the country had calculated its 2016 budget based on an oil price of $50 per barrel, a figure he said was an "optimistic" assessment of the situation.
Finance Minister Anton Siluanov said earlier this month that the budget would have to be adjusted to "new realities."
The government recently announced that it was preparing an anti-crisis plan to support sectors of the economy affected by the crisis.