The U.S. National Hurricane Center said the hurricane was 80 miles (130km) east of the port of Salina Cruz in Oaxaca state. Winds were blowing at a maximum 75 miles per hour (120kmh).
The hurricane was churning north-northeast at around 9mph (14kmh) and should start to weaken quickly by Wednesday night (local time).
"We are seeing very, very strong winds and intense rainfall," said Manuel Maza, director of emergency services in Oaxaca. He said power outages also were reported.
Mexican state oil monopoly Pemex said earlier on Wednesday operations were normal at its biggest refinery, which as the capacity to process 330,000 barrels per day of crude and is on the coast in Salina Cruz.
Local emergency services said a 61-year-old U.S. man surfing off the beach at Salina Cruz had drowned during the storm. A 26-year-old Mexican man was killed as he tried to cross a river.
Ports for small vessels in the area have been closed and emergency services in Oaxaca said they were starting to evacuate residents from some areas as a precaution, including the immediate vicinity of the refinery.
The NHC issued a hurricane warning from Oaxaca's Puerto Angel to Barra de Tonala, and a tropical storm warning from Barra de Tonala to Boca de Pijijiapan in Chiapas state.
Between 6 and 10 inches (10 to 20 cm) of rain is expected over eastern Oaxaca through western Chiapas, along with a storm surge of 3 to 5 feet (1 to 1.5 meters) above normal tide levels, the NHC said.
