At the same time, Conte pledged his new government would stick to European Union discipline rules in terms of public accounts and budget.
The lower Chamber of Deputies, where Conte's second government has a fairly comfortable majority, was set to vote Monday evening.
Conte, a law professor with no political affiliation, added: "The language of this government will be mild-mannered because we understand that our actions will not be judged by the arrogance of our words".
Conte lobbed his own barbs at Salvini, who abruptly withdrew the League as a partner in the previous government in a bid for an early election that would make him premier and bring him "full powers". Conte's fortunes are dicier in the Senate, where a coalition of the populist 5-Star Movement and center-left Democrats holds a much-slimmer margin in Tuesday's vote.
Luigi Di Maio, M5S leader and newly appointed foreign minister, said after the vote, "Now is the time for courage", while his PD counterpart, Nicola Zingaretti, said the confidence win marked "another step towards changing Italy".
"We also need to improve the Stability and Growth Pact and its application, to simplify the rules, avoid pro-cyclical effects, and support investments", Conte told parliament in reference to rules that restrict European government budgets, including in heavily indebted Italy. The firebrand League leader bet the move would trigger an early election to him the premiership. If he loses, he must resign as premier.
As interior minister in Conte's previous government, Salvini stiffened Italy's tough illegal immigration policies by banning migrant rescue boats run by charities from entering Italian ports.
Conte indicated that policy will be tweaked, without giving details.
One lawmaker from a tiny opposition party expressed ambivalence about the new coalition, still containing populists.
The PD and the 5 Stars, were bitterly fighting accurate by the previous year and can bear to work together within the event that they wish Mr Conte's second mandate to last. However, Riccardo Magi, from the More Europe party, announced reluctant support, saying the risk of an "illiberal and anti-European drift" if Conte's government falls would be worse.
The most pressing issue for the new coalition will by the upcoming 2020 budget, a key test for relations with Brussels.
Harsh EU fiscal rules, coupled with a perceived lack of EU solidarity as Italy struggled to serve huge numbers of migrants who headed to Europe by crossing the Mediterranean Sea, helped boost the popularity of Salvini's "Italians First" League.