A League source said the party would not block any quick political solution that would enable Italy to deal with possible "emergencies".
In the gilded Quirinale Palace, President Sergio Mattarella presided over the ceremony.
The two populist party leaders are both deputy prime ministers as well as key ministers in the new government.
"So far a crash in Italian debt markets have pulled populists back from the brink, and at present, their leaders are trying to form a less confrontational government".
In the end, the populist parties made a decision to back down and accept the president's veto of their original choice of a eurosceptic economy minister.
The ceremony capped a rollercoaster week of political and financial turmoil that saw stock markets around the world plunge and Italy's borrowing rates soar on the threat of a new election in Europe's third-largest economy.
Republic Day commemorates the day, June 2, 1946, when Italians voted in a referendum to abolish the monarchy in favor of a republic, Italy's first.
In a Facebook post, Di Maio gushed: "There are a lot of us, and we're ready to launch a government of change to improve the quality of life for all Italians".
Mr Conte is taking over from Paolo Gentiloni, who led a coalition cabinet for two years. In addition, Di Maio will run the Labor Ministry and Salvini the Interior Ministry, two of the most important portfolios in Italy today.
"I have never said that we should leave the euro", he added.
Yet migrant arrivals to Italy actually plunged in the previous year under the center-left Democratic Party, which signed controversial deals with Libya to beef up coastal patrols and prevent migrants from setting out in smugglers' boats across the Mediterranean Sea. "It is a European nation", he told Reuters.
A new migrant policy, which Salvini insisted upon during his campaign, is also likely to cause friction with Italy's European partners.
Another Cabinet pick seen as reassuring to those concerned the populists could set Italy drifting from its strong ties with the European Union is Foreign Minister Enzo Moavero Milanesi.
Salvini has branded as "racist" advice from European Commission President Jean-Claude Juncker for Italians to work harder and be less corrupt instead of blaming their woes on the EU.
The initial failure of Conte to form a government had alarmed financial markets, which feared a quick return to the polls that risked being tantamount to a plebiscite on Italy's keeping the euro currency.
"That means more work, less corruption". Seriousness", he said. "We will help them, as we always did.
Since Salvini took over the League, almost 700,000 people have landed in Italy after crossing the Mediterranean sparking a sense of resentment among many Italians who feel Europe has abandoned them - a sentiment which he has successfully exploited.
"I look forward to carrying on our close partnership with you and to continue deepening it", she said.
Offering the new government cautious support was Italy's small, far-right neo-fascist CasaPound party, which held its own Republic Day commemoration on Saturday. In Italy, opposition politicians see the new government contract as "dangerous" and "illusionist".