At 0725 GMT, the rouble was 1 percent stronger against the dollar at 64.12 and had gained 0.9 percent to trade at 71.61 versus the euro.
Brent crude oil, a global benchmark for Russia's main export, was up 1.1 percent at around $47.5 a barrel.
It hit a three-week high as speculation built about potential producer action to support prices.
Russian Energy Minister Alexander Novak said in an interview with the Asharq Al-Awsat newspaper that Russia was consulting with Saudi Arabia and other countries to achieve oil market stability.
The Russian currency gained around 1.2 percent against the dollar last week, but fell on Friday despite oil prices rising.
Dmitry Polevoy at ING Bank said in a note that the rouble should gain ground on Monday.
Polevoy said dividend payments could be behind Friday's fall in the rouble, while others speculated that investors were keen to trim rouble exposure over the weekend in light of a flare-up in tensions between Russia and Ukraine over Crimea.
Russian companies were due to pay insurance premium payments on Monday as the monthly tax period started.
The rouble is typically supported during that period as exporters sell a portion of their foreign-currency earnings to pay the rouble-denominated taxes.
Russian share indexes also moved higher on Monday.
The dollar-denominated RTS index was up 1.4 percent to 968 points, while the rouble-based MICEX was 0.2 percent higher at 1,970 points.