The 52-year-old, who is a relative of the child, then tried to kill herself.
The baby was found dead in an apartment in the Little Village neighborhood in Chicago at about 9.40 am on Monday morning, police told the Chicago Tribune.
She has been identified as Rose Herrera.
The woman had tried to silence the crying baby by shoving something, possibly a cloth, down her throat, a source told the Chicago Tribune.
When that didn’t work, she cut the baby’s throat with a chain saw, police said.