Ms Meloni, of the nationalist Brothers of Italy party, led a conservative alliance to triumph at Sunday's election.
Near final results showed the rightist bloc should have a solid majority in both houses of parliament.
Addressing cheering supporters, Ms Meloni said: "We must remember that we are not at the end point, we are at the starting point.
"It is from tomorrow that we must prove our worth."
Giorgia Meloni looks set to become Italy's first female PM
GUGLIELMO MANGIAPANE
Ms Meloni, of the nationalist Brothers of Italy party, led a conservative alliance
YARA NARDI
The 45-year-old faces soaring energy prices, war in Ukraine and renewed slowdown in the euro zone's third-largest economy.
Her coalition government, Italy's 68th since 1946, is unlikely to be installed before the end of October.
Prime minister Mario Draghi will remain at the head of a caretaker administration until a formal government is confirmed.
Initial market reaction on Monday was muted, with the outcome having been widely forecast in opinion polls.
Giovanni Donzelli, a key ally of Ms Meloni, told broadcasters: "What Italy needs is a stable government.
"The results appear to give us this possibility and we won't shy away from it."
The party is descended from Italy’s fascist movement and comes to power 100 years after Benito Mussolini and his Blackshirts marched on Rome.
But Ms Meloni has played down her party's post-fascist roots, portraying it as a mainstream group like the Conservative Party in the UK.
She has pledged to back Western policy on Ukraine and not take risks with Italy's fragile finances.
In her victory speech, she said: "If we are called on to govern this nation we will do it for all the Italians, with the aim of uniting the people and focusing on what unites us rather than what divides us.
"This is a time for being responsible."
With results counted in more than 97 percent of polling stations, the Brothers of Italy led with more than 26 percent, up from just four percent in the last national election in 2018.
The party supplants Matteo Salvini's League party as the driving force on the right.
The League took only around nine percent of the vote, down from more than 17 percent four years ago.
The other major conservative party, Silvio Berlusconi's Forza Italia, scored around eight percent.
Ms Meloni's alliance is split on some highly sensitive issues that might be difficult to reconcile once in government.
Mr Salvini, for example, questions the West's sanctions against Russia and both he and Mr Berlusconi have often expressed their admiration of Vladimir Putin.
They also have differing views on how to deal with surging energy bills and have laid out a raft of promises, including tax cuts and pension reform, that Italy will struggle to afford.
Ms Meloni will take over from Mr Draghi, the former head of the European Central Bank, who pushed Rome to the centre of EU policy-making during his 18-month stint in office, forging close ties with Paris and Berlin.
In Europe, the first to hail her victory were right-wing opposition parties in Spain and France, and Poland and Hungary's national conservative governments.