The S&P BSE Sensex index rose as much as 245.65 points touch 40,297.52 in the first few minutes of trade, and the broader NSE Nifty benchmark climbed to as high as 11,914.90, up 70.8 points from the previous close.
Gains in banking and information technology stocks pushed the markets higher. Analysts expected some volatility ahead of the expiration of monthly derivatives (futures & options) contracts due at the end of the session.
Equities in other Asian markets rose on Thursday and US stock futures edged higher after the US central bank cut interest rates as expected to keep economic expansion on track.
MSCI's broadest index of Asia-Pacific shares outside Japan was last seen up 0.2 per cent while Japan's Nikkei stock index rose 0.41 per cent, but Australian shares fell 0.24 per cent.
Meanwhile, the Sensex had ended 220.03 points - or 0.55 per cent - higher at 40,051.87 and the Nifty settled at 11,844.10, up 57.25 points (0.49 per cent) from its previous close on Wednesday, continuing their upmove for a fourth consecutive session.