The rouble gained 0.5% versus the dollar and more than 0.8% against the euro at market opening before retreating slightly.
As of 0716 GMT, the rouble was 0.3% stronger at 63.05 versus the dollar and gained 0.7% to trade at 71.36 versus the euro.
The rouble took advantage of globally improved risk sentiment after the United States and China agreed at the G20 summit on Saturday to restart trade talks.
"Risk-assets are broadly starting the week on a stronger footing amid positive signals from the Trump-Xi meeting on Saturday: both sides agreed to continue trade talks," VTB Capital said in a note.
President Vladimir Putin, who also attended the summit in Osaka, supported the rouble and Russian stocks by saying that Russia and Saudi Arabia have agreed to extend oil output cut deal, daye before the OPEC meeting in Vienna.
Brent crude oil, a global benchmark for Russia's main export, jumped 2.8% higher to $66.57 a barrel.
The rouble-based MOEX Russian index hit a fresh record high of 2,793.02 before sliding to 2,790.3, up 0.9% on the day. Its dollar-denominated peer RTS was up 1% at 1,394.3 points.