A record-setting spree on local bourses too bolstered sentiment on the forex trading front.
This is the highest finish for the home currency since April 29, 2015 when it had settled at 63.30.
Overall forex sentiment got a leg-up ahead of the advanced gross domestic product (GDP) growth estimates for 2017-18, which was released after the market hours.
The projections are very crucial in view of upcoming Union Budget and also the first official full-year growth estimates forecast after the GST regime.
Economic activity regained its footing in the second quarter of the current fiscal after GDP rebounded to 6.3 per cent July-September period, sending out a comfort signals about bottoming out of economic slowdown.
The Indian rupee has strengthened by 55 paise since the start of 2018.
Brent crude, an international benchmark, is trading marginally down at USD 67.72 a barrel in early Asian trade.
In the meantime, domestic equity markets continued their new year party with both the flagship indices conquering new highs on persistently supportive liquidity driven rally.
Buoyant global cues too supported the upbeat trend as most Asian bourses had a great tailwind session after Dow Jones closed above the 25,000-mark.
Sensex shot up over 184 points to finish at 34,153.85, while Nifty jumped 54 points to 10,558.85.
Extending its bullish momentum, the rupee opened on a high note at 63.36 a dollar against Thursday's close of 63.41 at the Interbank Foreign Exchange (forex) market on sustained selling of the American currency by exporters and banks.
A strong follow-through traction, later pushed the local unit decisively to hit an intra-day high of 63.31 in mid-morning deals before suddenly reversing the trend to touch a fresh low of 63.42 for a brief period.
But, later it bounced back to close comfortably at a new 32-month high of 63.37, showing a gain of 4 paise, or 0.06 per cent.
For the first week of 2018, the rupee has appreciated by a healthy 50 paise.
The dollar index, which measures the greenback's value against a basket of six major currencies, was up at 91.75 in early trade.
Globally, the greenback remained higher against other major currencies ahead of the highly-anticipated US jobs data due later in the day.
In cross-currency trades, the rupee firmed up further against the pound sterling to end at 85.81 per pound from 85.89 and the Japanese yen rose sharply to finish at 55.96 per 100 yens from 56.31 earlier.
The home unit also recovered against the euro to close at 76.37 compared to 76.39 yesterday.
Elsewhere, the common currency euro loitered near its three-year peaks against the US dollar.
In forward market today, premium for dollar edged lower due to mild receiving from exporters.
The benchmark six-month premium payable in June eased to 133-135 paise from 134.50-136.50 paise and the far forward December 2018 contract also edged down to 269-271 paise from 270.50-272.50 paise.