Short covering due to expiry of April series F&O contracts added to the momentum, traders said.
After surging 1,167 points during the day, the 30-share index settled 997.46 points or 3.05 per cent higher at 33,717.62. It hit an intra-day high of 33,887.25.
Similarly, the NSE Nifty soared 306.55 points, or 3.21 per cent, to 9,859.90.
ONGC was the top gainer in the Sensex pack, rallying over 13 per cent, followed by HCL Tech, Hero MotoCorp, NTPC, TCS, M&M, Infosys and Maruti.
Shares of Reliance Industries and Tech Mahindra climbed around 3 per cent ahead of their earnings announcements, while HUL settled in the red.
Sun Pharma, IndusInd Bank and Asian Paints too closed with losses.
Most major global markets have rallied in the current week as several countries have started to talk about lifting the lockdown, said Sanjeev Zarbade, VP PCG Research, Kotak Securities.
"Reports of encouraging results about Gilead's drug in treating COVID-19 as well as starting of human trials on a vaccine also fuelled the rally," he said.
Top US epidemiologist Anthony Fauci said that Gilead Science's remdesivir "has a clear-cut, significant, positive effect in diminishing the time to recovery".
Bourses in Shanghai and Tokyo ended with significant gains, while those in Hong Kong and Seoul were closed for a holiday.
However, stock indices in Europe were trading on a tepid note in early deals.
International oil benchmark Brent crude futures rallied 9 per cent to USD 26.40 per barrel.
Further, rising buzz of another stimulus package by the government, focussed on industries, kept investor sentiment positive, traders said.
On the currency front, the rupee surged 57 paise to provisionally close at 75.09 against the US dollar.
Meanwhile, global tally of coronavirus infections climbed to 31.93 lakh, with around 2.27 lakh deaths.
The death toll due to COVID-19 in India rose to 1,074 and the number of cases climbed to 33,050, according to the union health ministry. ANS ABMABM