Storm-force winds caused minor damage in the Manu'a Islands group of American Samoa.
Red Cross officials said blustry winds nearly blew down some of the tents in which survivors of the Sept. 29 quake and tsunami are still living.
The Cook Islands issued a cyclone alert in anticipation of destructive winds of up to 70 mph when Nisha arrived late in the week.
The Hawaii-based U.S. military Joint Typhoon Warning Center said that the cyclone was expected to track across the Cook Islands before aiming toward southern parts of French Polynesia.
Cyclone Nisha Track
Satellite Loop Data: CIMSS