Noda, 54, who was finance minister in the Kan Cabinet, had earlier been elected president of the ruling Democratic Party of Japan in a runoff, clearing the way for him to become the prime minister under Japanese practice.
Earlier Tuesday, Kan and his Cabinet resigned en masse to enable the new prime minister to form his government later this week, Kyodo News reported.
Kan, who held office only since June of last year, stepped down last week after coming under criticism for his government's handling of the nuclear crisis triggered by Japan's March 11 earthquake and tsunami. The disaster caused hundreds of billions of dollars in damage to the Japanese economy.
Among the tough challenges facing Noda are rebuilding the country and unification of his divided party, the BBC said.
Noda is seen as a fiscal conservative who has supported increasing taxes, such as doubling Japan's 5 percent sales tax to fund the recovery, reduce debt and meet the country's social-security costs.
The BBC said Noda wants to restart nuclear reactors idled after the disaster at the Fukushima Daiichi nuclear plant.
"Running Japan's government is like pushing a giant snowball up a snowy, slippery hill," Noda had said earlier, CNN reported. "In times like this, we can't say, 'I don't like this person' or 'I don't like that person.' The snowball will slide down."
An analyst told CNN the country's political problems, as witnessed by the succession of prime ministers, are affecting the economy, which is the world's third largest after the United States and China.
Moody's rating agency last week downgraded Japan's rating a notch to Aa3 from Aa2, citing its huge deficit and frequent changes in the administration.