However, broader gains were capped as domestic shares fell for a third consecutive session, retreating from record highs hit on Friday, as momentum waned ahead of the conclusion of general elections next month.
In the latest large opinion poll, the opposition Bharatiya Janata Party and its allies were forecast to win a narrow majority in the 543-seat lower house of parliament, compared with previous surveys predicting that they would fall short.
"Until May 16, we will trade between 59.60 and 61.30 as market is positioning for rupee appreciation given the opinion polls outcome, and FII (foreign institutional investor) flows," said Hemal Doshi, a currency strategist at Geojit Comtrade Ltd.
The partially convertible rupee closed at 60.42/43 per dollar compared with 60.6450/65 on Monday.
Large corporate dollar sales helped offset continued month-end demand for the greenback from oil companies.
Global investors are largely awaiting the outcome of the two-day Federal Reserve's policy meeting on Wednesday, with the U.S. central bank widely expected to continue paring back its bond-buying monetary stimulus.
The South Korean won hit a near six-year high on Tuesday, leading gains among emerging Asian currencies, though many investors stayed cautious ahead of major events this week including the Fed policy meeting.
In the offshore non-deliverable forwards, the one-month contract was at 60.74 while the three-month was at 61.53.
FACTORS TO WATCH
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Foreign institutional investor flows
For data on currency futures