The 19-year-old British No1 was beaten in 86 minutes by her 43-year-old opponent, who is currently ranked 12 places below Robson.
Robson, who was the seventh seed at the event, struggled with her second serve as Date-Krumm – 24 years her senior – broke twice in the opening set.
Date-Krumm traded breaks with Robson in the second set, before striking again in the ninth game to book her place in the last 16.
Home favourite Date-Krumm will face fellow Japanese Misaki Doi in the next round for a spot in the quarter-finals.
Robson, who reached the last 16 at Wimbledon earlier this year, has won only 11 matches on the WTA Tour all season.
Defending champion Watson lost to Monica Puig in the first round in Osaka, going down 6-4 6-4 to the Puerto Rican.
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