Ms Meloni is widely expected to form Italy's most right-wing government since World War Two.
That will alarm much of Europe as Italy is the EU's third-biggest economy.
However, speaking after the vote, Ms Meloni said her Brothers of Italy party would "govern for everyone" and would not betray people's trust.
"Italians have sent a clear message in favour of a right-wing government led by Brothers of Italy," she told reporters in Rome.
She is predicted to win between 22-26% of the vote, says a Rai exit poll, ahead of her closest rival Enrico Letta from the centre left.
That dominance was underlined by the first projection from Rai, which gave her more than a quarter of the vote for the Senate. Projections are based on concrete results. Italians also vote for the lower house, the Chamber of Deputies.
Ms Meloni's right-wing alliance - which also includes Matteo Salvini's far-right League and former PM Silvio Berlusconi's centre-right Forza Italia - now looks to have control of both houses, with a projected 42.2% of the Senate vote.
But the decision on who becomes Italy's next leader is up to the president, not Giorgia Meloni, and that will take time.
Although she has worked hard to soften her image, emphasising her support for Ukraine and diluting anti-EU rhetoric, she leads a party rooted in a post-war movement that rose out of dictator Benito Mussolini's fascists.
Earlier this year she outlined her priorities in a raucous speech to Spain's far-right Vox party: "Yes to the natural family, no to the LGBT lobby, yes to sexual identity, no to gender ideology... no to Islamist violence, yes to secure borders, no to mass migration... no to big international finance... no to the bureaucrats of Brussels!"