Cops fatally shot a knife-wielding woman near her Midtown homeless shelter after she chased after another resident with the blade, authorities said.

The 57-year-old woman, whose name wasn’t released, began fighting with the other woman at around 8:30 p.m. in the New Providence Women’s Shelter on East 45th Street near Second Avenue, cops said.

The fight spilled outside, where two cops ordered her to drop the dagger, but she started coming at them.

“The police said, ‘Put the knife down,’ and ‘Get on the floor,’ ’’ shelter resident Betty Saunders said. “They even said, ‘Please.’ ”