The authorities said the woman had flown to Costa Rica from Peru.
She was arrested in a hotel in the Costa Rica's capital, San Jose.
Honduran officials said they believed that the five Syrians held on Tuesday were refugees, and there was no evidence they were Islamist militants.
The five were detained when it became clear they were carrying stolen Greek passports. They told police that four of them were students and one a professional worker.
A sixth man was turned away from Honduras on Friday on arrival by plane from El Salvador, and was sent back.
In Paraguay, police detained a seventh Syrian man on Sunday who was also travelling on a stolen Greek passport.
Migration experts say that for several years migrants from Iraq, Afghanistan, Syria and other countries have tried to enter the US through Central America and Mexico.
But news of the Syrians' arrests is sparking alarm in the US, where some fear that militants may use the same smuggling routes.
Costa Rica has detained a Syrian woman with a fake Greek passport, two days after five Syrian men were arrested in Honduras trying to travel to the US on stolen passports.