Gentiloni, 62, was asked by President Sergio Mattarella on Sunday to form a new centre-left government that will guide Italy to elections due by February 2018, following the resignation of outgoing Prime Minister Matteo Renzi.
Opposition parties slammed the softly-spoken former foreign minister as little more than a Renzi puppet, but Milan's FTSE Mib saluted the move, up 1.04 percent at 2:00 pm (1300 GMT).
It was also buoyed by relief over the news the Italian government would intervene to recapitalize Italy's Monte dei Paschi di Siena bank (BMPS), should it fail to raise the money from private investors needed to stay afloat.
Silver-haired Gentiloni, a one-time student radical from an aristocratic family, is expected to keep the cabinet largely untouched and present his final list to the president by the close of play Monday.
He will then seek parliamentary approval of his new government on Tuesday or Wednesday.
The biggest cabinet seat to fill is the one left vacant by Gentiloni himself, that of foreign minister.
Political watchers say it could go to Piero Fassino, a member of Renzi's centre-left Democratic Party (PD) who has previously held the justice and foreign commerce portfolios.
Interior Minister Angelino Alfano, who was Renzi's deputy and heads the New Centre-Right (NCD) party, is also tipped for the post.
Should it go to Alfano, the interior portfolio could be handed to Domenico Minniti, the state secretary with responsibility for the security services under Renzi.
Analysts say Gentiloni could also keep the foreign minister job for himself, at least in the short term. Pier Carlo Padoan is expected to stay on as finance minister to reassure Europe that the eurozone's third-largest economy is on solid ground.
Among the most pressing issues facing the new government is the fate of the troubled BMPS.
The institution, the third largest in Italy, had requested extra time from Europe to plug a gaping hole in its finances, but reports on Friday that the European Central Bank had refused spooked the markets.
But just hours after Gentiloni was named as prime minister, the bank said it could avoid appealing for a government bailout, with BMPS shares up 6.46 percent in early afternoon trading on Monday.
Oanda analyst Craig Erlam said investors were “more optimistic” the bank could raise the 5 billion euros ($5.29 billion) needed to avoid a handout and were relieved political uncertainty had been removed in the short term at least.
Gentiloni is now rushing to resolve the political crisis sparked by Renzi's crushing referendum defeat and downfall in time for Italy to attend the European Council meeting in Brussels on Thursday, where the pressing issue of migration is on the table.
Italy is on the frontlines of the migrant crisis, with a record 175,000 people landing on its shores this year alone.
“The Gentiloni-Padoan government is coming together at a super speed to prevent the implosion of the Siena bank (BMPS) and to make sure Italy does not turn up at the European Council with an incomplete government,” La Stampa daily said.
Renzi may be down and out for now, but analysts said he had tapped Gentiloni to replace him because he trusts him to keep his seat warm for the next general elections, which could be brought forward to early next year.
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By Ella Ide