An online vote on whether to take part was delayed earlier this week when M5S founder Beppe Grillo asked Draghi for more details on action he would take on the environment.
The post of justice minister has been given to Marta Cartabia (independent), of defense minister to Lorenzo Guerini (Democratic Party), of economy and finance to Daniele Franco (Bank of Italy director general), of economic development to Giancarlo Giorgetti (Lega), of agriculture policies to Stefano Patuanelli (M5S, he was responsible for economic development in the previous government), of ecological transition to Roberto Cingolani (independent), of transport and infrastructure to Enrico Giovannini (independent), of labor and social policies to Andrea Orlando (PD), of education to Patrizio Bianchi (independent), of universities and research to Cristina Messa (independent).
Mr Draghi, 73, replaces Giuseppe Conte, who resigned after a small party pulled support over the handling of the pandemic.
Members of Italy's 5-Star Movement voted on Thursday to back Prime Minister designate Mario Draghi, opening the way for the former European Central Bank chief to take office at the head of a broad government of national unity.
More than 93,000 people with coronavirus have died in Italy since it became the first European country to face the full force of the pandemic one year ago, and the toll is still rising by the hundreds each day.
Draghi is due to return to the president's office on Saturday at midday to be sworn in.
Lawmakers have since failed to form a new alliance.
The anti-establishment Five Star Movement's Luigi Di Maio is expected to remain in place as Italian Foreign Minister.
He has secured the support of nearly all the main political parties, following the collapse of the previous administration last month. Draghi also has the support of former Premier Silvio Berlusconi's center-right Forza Italia and former Premier Matteo Renzi's Italy Alive Party.
His arrival was greeted with delight by the financial markets, and Italy's borrowing costs dropped to a historic low this week.
Draghi has also said he will make the anti-coronavirus vaccine program a priority.
The economy shrank by a staggering 8.9 percent previous year, while Covid-19 remains rife and restrictions including a night curfew and the closure of bars and restaurants in the evening remain in place.
In one of the last acts of his government Friday, Conte's cabinet extended a ban on travelling between regions for another week, and tightened curbs in four regions.
The country is pinning its hopes on receiving more than 220 billion euros ($267 billion) in European Union recovery funds to help get back on its feet.
The country is at a critical juncture as it battles both the medical and economic consequences of the coronavirus pandemic, which struck Italy first outside of Asia nearly exactly one year ago.
But Draghi will have to balance demands for immediate hand-outs against the need for long-term structural reforms in Italy - tensions that brought down the last government.