It happened on Walnut Avenue and Linwood Street shortly after 7:30 pm.
Police say the woman was walking her one-year-old child in a stroller on the sidewalk when a white Nissan veered off the road and hit them.
The driver, who officers identified as Leonel Hernandez Alacon, took off without stopping before abandoning the car nearby.
Officers arrested Alacon as he was trying to run away.
He's been booked in the Tulare County jail on several DUI and hit-and-run charges.
The victims were evaluated and released by paramedics.