"The Indian rupee has got its mojo back to a certain degree, part of it reflects international factors, (but) one of the key factors has been the soothing impact of (Reserve Bank of India) governor Rajan's appointment," Nick Verdi, director, FX Strategy Asia Pacific ex-Japan, Barclays told CNBC.
"He's had a Draghi-like impact on the rupee, promising measures if the rupee weakens from here," he said.
A string of positive economic indicators released in recent days is fueling optimism that the economy is gaining momentum after three straight quarters of sub-5 percent growth, economists said.
The country's trade deficit, for example, narrowed to $10.9 billion in August , from $12.27 billion in July, helped by a double-digit rise in exports. Exports increased by 13 percent, the second consecutive double-digit rise, helped by an improvement in overseas demand. Separately, industrial output unexpectedly rebounded in July, rising 2.6 percent from a year earlier, following a 2.2 percent contraction in June.
(Read more: RBI's Rajan takes a deep dive to save the rupee)
"Looking ahead, we have argued that the combination of a pick-up in the world trade cycle and the lagged effects of rupee depreciation, which started in mid-2011, will see Indian export growth recover more strongly than generally expected, pushing industrial activity up with it," said Robert Prior-Wandesforde, head, India and South East Asia Economics at Credit Suisse.
Discussing the change of mood over India, Prior-Wandesforde cautioned, while a "modicum of calm" has been regained, this could reverse as quickly as it came.
Reflecting uncertainty over the economy's outlook, India's wholesale price index (WPI), published on Monday, rose at its fastest pace in six months during August. It gained 6.1 percent, compared with expectations for a rise of 5.8 percent.
"We're still in dangerous, nervy territory," he said, pointing to a host of risk events including the Federal Open Market Committee meeting on September 17 and 18, and the Reserve Bank of India's mid-quarter policy review on September 20.
(Read more: Mobius: The rupee rout will be over soon)
For international investors, it is vital that RBI governor Rajan has clear objectives and adopts a consistent communication strategy, say experts.
"In Rajan's first speech, the tone of the RBI's likely policy goals was outlined, but actions will speak louder than words in the coming months. If actual policy does not follow the outlined objectives then the RBI could face market skepticism about its credibility," said Paul Mackel, head of Asian FX Research at HSBC.
—By CNBC's Ansuya Harjani; Follow her on Twitter @Ansuya_H