Dolly moved ashore in the evening south of Tampico port in the state of Tamaulipas, blowing maximum sustained winds of 75 kilometers per hour (45 miles per hour), according to the US National Hurricane Center.
The storm was expected to produce between 150 and 250 millimeters (five and 10 inches) of rainfall in Tamaulipas and the neighboring states of Nuevo Leon, Veracruz and San Luis Potosi.
The downpour threatened to trigger flash floods and mudslides in mountainous regions.
The government of Tamaulipas prepared shelters, while Veracruz authorities said classes would be suspended on Wednesday.
On the other coast, Tropical Storm Norbert swirled along the Pacific, about 185 kilometers from Cabo Corriente. It was forecast to gain strength, with top winds of 75 kilometers per hour, the US Hurricane Center said.
But it was expected to move away from the southwestern coast overnight and approach the southern tip of the Baja California peninsula on Thursday.
The storm is expected to pour 50 to 100 millimeters of rainfall over the states of Colima, Michoacan and Jalisco.
Last year, Mexico was simultaneously struck by a pair of hurricanes, Ingrid and Manuel, on both coasts, killing 157 people, destroying bridges and burying most of a mountain village in the Pacific coast state of Guerrero.