The 22-year-old Morita, ranked 83rd in the world, secured a 6-4, 6-4 win over teenager Alison van Uytvanck to put Japan one up early in the day.
Veteran Date-Krum, former world number four, then defeated Tamaryn Hendler 6-1, 6-4 to move closer to the elite eight-nation World Group in next's year Fed Cup.
The losers of the tie will be relegated to World Group II.
Belgium missed former world number one Kim Clijsters, who has a torn hip muscle, and the country's highest-ranked player Yanina Wickmayer, who wanted to concentrate on the WTA Tour.
Sunday will see two reverse singles matches and one doubles match with Morita teaming up with Rika Fujiwara against Ysaline Bonaventure and An-Sophie Mestach. AFP