Presented in October, the budget was approved by the upper house on the night of Thursday into Friday with 215 votes, 16 against and no abstentions.
It now has to be passed by the Chamber of Deputies before the end of the year.
The main beneficiaries "are salaried people and pensioners with little incomes and means," Draghi, a former chief of the European Central Bank, told reporters.
Italy was the first EU country to experience a major outbreak of Covid-19 in early 2020.
A whopping 32 billion euros ($36 billion) has been set aside to kickstart the economy and support struggling businesses and families amid a surge in coronavirus cases.
The income tax brackets have been reduced to four categories from the current five with fiscal waivers for those with lower salaries.
The biggest benefits will be for people in the third bracket -- or drawing annual salaries of between 28,000 euros and 50,000 euros -- whose taxation rates will fall to 35 percent from 38 percent.
Conversely, the tax rates for people drawing 50,000 euros and more will rise from 41 percent to 43 percent, or the highest rate, which was earlier applicable for incomes exceeding 75,000 euros.
The budget also exempted some 835,000 entrepreneurs from paying a regional tax for independent businesses and set aside 3.8 billion euros in 2022 to help households cope with higher energy costs.
"We are ready to add supplementary resources if the price spiral does not stabilise," said Draghi.
The Italian leader said this week the country "has achieved all 51 targets" needed to qualify for the next tranche of nearly 200 billion euros allocated to Italy as part of the EU's post-virus fund.
Brussels has demanded reforms in return for the grants and loans, including an overhaul of the justice system, one of the least efficient in Europe, and a reform aimed at regularising real estate.
Rome received its first check in August for 24.9 billion euros from the European Commission -- 13 percent of the total funds expected from Brussels over six years.