World number 38 Robson and Konta, ranked 125th, both reached the quarter finals in Guangzhou, China, last week, but there were contrasting fortunes in their next tournaments.
In Japan, both British number one Robson and world number 62 Morita broke serve twice in a fraught first set, but the Japanese player prevailed in the tie-break.
The 23-year-old Morita capitalised on her only break point of the second set to wrap up a straight sets win and progress to the next round where she will face Jelena Jankovic.
Konta, meanwhile, stayed in China but returned to the second tier of women's tennis after her exploits at Guangzhou.
The British number three blasted nine aces en route to an epic 6-4 4-6 6-3 victory over eighth seed Lauren Davis of the United states in round one of the WTA 125 event at Ningbo.
The match lasted over three hours, with Konta, 22, facing Katarzyna Piter in the second round.