The infant girl was not injured and had been turned over to her father, the San Francisco Chronicle reported. The dog was with city animal care and control.
Police officer Tracy Turner said the 35-year-old woman, who had not been identified, was struck by the city Recreation and Parks vehicle in Holly Park about 2:20 p.m. PDT, the newspaper said.
The unidentified driver, who has worked for the department since 2009, left the park but was taken into custody by police several blocks away. Turner said police planned to book him on suspicion of felony hit and run.