The stationary vehicle, a large BMW saloon, had been parked up at the time before it was slammed into the buggy.
The incident happened in a Cork city suburb of Blackrock on Wednesday afternoon.
The child was taken to Cork University Hospital, where it was kept in for observation.
However, a garda spokesman said the child was fine and that two people had been arrested in connection with the incident.
It happened at around 3pm on the Blackrock Road on the southside of the city in a location which is close to a busy national school.
It is understood that the driver of the offending car may have earlier been involved in an accident in the city centre.
Lord Mayor of Cork and local councillor Des Cahill said he was just glad there was no serious injuries.
“It is an horrific incident which could have involved a lot more people,” he said. “It is a very pedestrianised area and would have felt like a very safe area.”
However, he added that there are very few sets of traffic lights along the Blackrock Road and that meant it was a road that could lead to speed. Cllr Cahill said he had been calling for some time for more traffic measures to be put in place but they just had not been forthcoming.