Japan's Self-Defence Forces were deployed to Nagano prefecture to help with search and rescue operations, public broadcaster NHK reported.
The typhoon brought powerful winds and heavy rain causing about 200 rivers to overflow. Levees burst on about 50 of them, causing flooding across widespread areas.
According to NHK, more than 10,000 houses have been damaged.
Fukushima prefecture was one of the hardest hit areas. Of the 66 victims, 25 people died in the prefecture.
Officials said that many areas in Fukushima received up to 40 per cent of their annual rainfall over a span of two days.
The heavy rain also caused about 140 landslides across the country. In Gunma prefecture, four people were killed when their homes were washed away.
Nearly 35,000 households were still without electricity. Another 130,000 homes have no running water as of Tuesday morning and it remained unclear when the utilities would be fully restored.
Meanwhile, Prime Minister Shinzo Abe on Tuesday said his government was planning to designate Hagibis a "severe natural disaster", in order to increase state subsidies for reconstruction work in affected regions, NHK reported.
Abe said the government was also exploring the possibility of making 315 municipalities eligible for "severe disaster" provisions, to help authorities carry out urgent repair work without financial concern.
( With inputs from IANS )