Rivers overflowed and rising floodwaters were three feet (1 metre) deep in coastal towns near the port city of Veracruz as the storm made landfall on Tuesday morning. There were no reports of people being injured or missing. Mexico's main oil export ports remained open and state oil company Pemex said output from only one natural gas field had been affected.
Four production platforms near the coast were evacuated on Monday afternoon but the country's giant Cantarell and Ku Maloob Zaap oil fields were unaffected.
Many coastal residents were scrambling to move their possessions and animals away from the flood plains.
There was no immediate word of any damage to the area's coffee and sugar crops.
Off Mexico's Pacific coast, Hurricane Norbert churned north towards the Baja California peninsula, where it was expected to make landfall on the weekend.
(Reporting by Manuel Carrillo and Robert Campbell; Editing by Eric Walsh)