The partially convertible rupee finished at 45.91/92 per dollar, weaker than Friday's 45.53/54 and its lowest close since September 29, 2004, when it ended at 46.00/46.01.
"We are seeing a bearish trend for the rupee. If the euro and yen do not recover tomorrow, then we could see the rupee depreciating further," said a dealer at a state-run bank.
The dollar spiked to a 26-month peak against the yen and an 18-month high versus the euro on Monday as investors expected the US currency's interest rate advantage to keep widening.
The dollar was around 117.74 yen after climbing to 118.38 yen, its highest level since August 2003.
The euro fell as low as $1.1783, an 18-month low, before tracking back slightly to $1.1818, flat on the day.
Traders took some comfort from a rise in the domestic stock market. India's main stock index ended up 1.7 percent on Monday after gaining 5 percent in previous three sessions.
The index fell steeply in October as foreign funds, wary of a rising dollar, pulled money out of Indian stocks.
Latest data showed overseas portfolio investors bought $87.6 million of Indian shares last Wednesday, before the market shut for a two-day holiday.
Foreign funds sold shares worth nearly $800 million in October, after buying a net $8.5 billion in the year to September.