The deaths and the worst damage occurred in the state of Veracruz, where the storm uprooted trees when it made landfall early on Saturday.
In the state capital, Xalapa, many streets became rivers of mud.
The hurricane later weakened to a tropical storm as it moved inland.
However, high winds and downpours were reported to be causing more flooding as Grace travelled north of Mexico City.
- Above-average hurricane season expected
- What are hurricanes, cyclones and typhoons?
The storm brought wind speeds of up to 200 km/h (125mph) when it reached the coast of mainland Mexico. Of the eight people killed, six were members of the same family, state government officials said.
Residents of the coastal town of Tecolutla, in Veracruz state, said the hurricane had been devastating.
“You could just hear the thunder of the falling trees, it sounded very nasty, very horrible,” Laura Jacinto said, adding: ” I was very scared, I was very scared last night.”
Another resident, Adolfo Lopez, said: “The roof… everything is gone, it took everything with it.”