The domestic currency has gained by four paise or 0.06 per cent in two days.
The rupee extended gains for the second straight session against the US currency by firming up 21 paise to close at 64.82 a dollar.
RBI fixed the reference rate for the dollar at 66.2930 and euro at 74.8978.
The rupee showed range-bound movement on Wednesday as investors preferred to stay cautious in the unsure market.
Sharp fall in domestic stock market also affected the rupee sentiment.
The rupee had slumped to its all-time closing low of 68.80 a dollar on August 28, 2013.
The rupee continued to slide against the pound sterling and finished at 102.64 as against 102.25 previously.
At the Interbank Foreign Exchange market, the domestic unit commenced higher at 61.80 a dollar from previous close of 61.92.
The rupee hit a near 10-month high as an alliance led by pro-reform and business friendly Hindu nationalist Narendra was on course for an absolute majority.
