"Soldiers fired on the vehicle after the driver failed to obey verbal and visual instructions to stop, switched off the vehicle lights, and forced guards out of the way as he attempted to bypass the checkpoint," a US military statement said on Tuesday.
It said the incident happened late on Monday.
The statement said the mother was taken to hospital while the father, who was driving, was questioned by police.
Hard to spot
Many Iraqis accuse US soldiers of being too hasty to open fire and of killing many innocent civilians.
Baghdad residents say they often fail to notice poorly lit and unannounced checkpoints while driving at night. Power cuts regularly plunge streets into darkness, making checkpoints even harder to spot.
The US military says soldiers need to defend themselves if they believe a checkpoint is under attack.