The incident happened at 9th Street and Fifth Avenue in the Park Slope section around 12:30 p.m.
Authorities say the young victims were a 4-year-old girl and a 1-year-old boy.
The little girl's 34-year-old mother was also struck by the vehicle. Witnesses say she is pregnant and was pushing the little boy in a stroller and carrying her daughter at the time.
She was taken to the hospital in critical condition, but is now stable and expected to survive.
Officials say another 34-year-old woman was also struck and is said to be stable.
"We do not exactly know what happened yet, there is a full investigation underway. But I will state the obvious, this loss of life is tragic, painful for all of us, particularly those of us who are parents," NYC Mayor Bill de Blasio said.
After striking the victims, the 44-year-old driver crossed into an oncoming lane and crashed into a parked vehicle, officials say. She was taken into custody.
Kemberly Richardson tweeted exclusive video of the incident: