The rupee climbed to 65.75 against the dollar, gaining nearly two percent from its previous close, on investor hopes the worst could be over for the currency, the worst performing in Asia this year.
Indian shares jumped as much as 2.96 percent at the open, led by banking stocks, after Rajan took over Wednesday from Duvvuri Subbarao as head of the Reserve Bank of India (RBI).
In the afternoon stocks were up 1.51 percent.
Rajan sought to reassure rattled markets with his first speech in the post, outlining a fresh approach to the currency crisis and warning that he may have to take unpopular steps to get Asia's third largest economy back on track.
Sonal Varma, an economist at Nomura Securities, said Rajan had made "an impressive start" but she stressed that a weak growth outlook was still a "major concern".
"In our view, amid the current gloom, the new RBI governor has infused a sense of optimism that he is in charge and that the RBI under him will unleash more financial sector ref