NHK said one man was found dead inside his car, which was submerged to the rooftop in the middle of a field, while another man died after being caught in a landslide.
One other person remains missing, and at least 87 people have been injured, NHK said.
About 340,000 households, most of them in Kyushu, were without electricity early on Monday, the trade ministry said, while Kyushu Railway, said it had halted operations on Kyushu and Japan Airlines and ANA Holdings cancelled about 800 flights, public broadcaster NHK reported.
The storm made landfall again in Shimane prefecture in western Honshu after tracking the coastline earlier on Monday, and was heading east at about 35kmh, the JMA said.
The storm will veer into the Japan Sea for a second time and track the coast to the north of Honshu into Tuesday before crossing overland and moving northeast out into the Pacific, the agency projected.
Up to 400mm of rain was expected in central Japan’s Tokai region, the nation’s industrial heartland, over the next 24 hours, it said.
Toyota Motor was among manufacturers that said they would idle production at some factories due to the storm, but there were no reports of major damage to industry.
Intermittent bouts of heavy rain lashed Tokyo but businesses in the capital were largely operating as normal.
Most schools were closed on Monday anyway for a public holiday.