The storm was located about 80 miles (125 kilometers) south-southeast of Punta Cana, Dominican Republic, and was moving northwest at 9 mph (15 kph) with winds of up to 35 mph (55 kph). It is expected to weaken further in the next two days.
Puerto Rico Gov. Alejandro Garcia Padilla activated the National Guard, canceled classes at all public schools and closed central government agencies as a precaution.
Heavy rains were still expected across Puerto Rico, according to state meteorologist Ernesto Morales.
"We should not lower our guard," he said. "The storm is very dangerous. There's a very high possibility of flooding."
Falling trees brought down a telephone post in the central town of Ciales, while officials reported small landslides in the central mountain town of Utuado and the southeast town of Yabucoa, with crews working to clear roads. Minor flooding was reported in the island's southeast region.
Several residents in the northern municipality of Bayamon were relocated because a nearby mountain had already experienced landslides in recent weeks amid persistent wet weather.
The storm was expected to drop up to 8 inches (20 centimeters) of rain in central Puerto Rico, and up to 4 inches (10 centimeters) elsewhere in Puerto Rico as well as in the U.S. Virgin Islands and eastern portions of the Dominican Republic. Officials in the U.S. Virgin Islands closed all schools in St. Croix.
Miguel Rios, director of Puerto Rico's emergency management agency, said the tropical storm was interacting with a cluster of storms hovering above Puerto Rico's northeast coast.
"In the next 36 hours, we can experience heavy rains at any moment," he said. "We must remain on alert."
Some 14 inches (36 centimeters) drenched the capital of San Juan in July, making it the wettest July ever recorded for Puerto Rico.
The National Hurricane Center in Miami has issued a tropical storm warning for parts of the Dominican Republic, from Cabo Engano to Cabo Frances Viejo.
According to the National Hurricane Center, all coastal warnings and watches have been cancelled for Puerto Rico. The government of the Dominican Republic has changed the tropical storm warning from Cabo Engano to Cabo Frances Viejo to a tropical storm watch, and has discontinued the tropical storm watch from Santo Domingo to Cabo Engano.