Stocks also gained as the rupee rose against the dollar, although profit booking at higher levels dragged the index below the 21,000 level.
ICICI Bank and HDFC Bank shares together contributed 163 points to the gains on the Sensex, which was also lifted by Larsen & Toubro and Reliance Industries.
BHEL and Maruti Suzuki were among the 20 index shares that moved up. Nine of the 13 BSE sectoral indices advanced, with a strong performance in bank and capital goods stocks.
The index is at the highest closing level since 20,974.79 on November 5.
The CNX Nifty on the National Stock Exchange spurted 80.15 points, or 1.3 per cent, to 6,241.10. The SX40 on the MCX Stock Exchange rose 120.07 points to 12,427.14.
Most exit polls indicated gains for the BJP in Delhi, Chattisgarh, Madhya Pradesh and Rajasthan. The rupee strengthened to 61.53 against the dollar in intra-day trade.
The election results will be announced on December 8. Parties that win the state polls may have a stronger footing during the Lok Sabha elections scheduled to be held by May.
"We believe that a stable government following the 2014 general elections could remove one key source of uncertainty and support a more sustainable growth uptick," Nomura said, noting the exit polls may differ from the official results.
