The crash, at 94th Street and Ditmars Boulevard, happened just after 9 a.m., authorities said, when the livery cab ran into the Corrections bus.
The bus was ferrying two prisoners from New York to court appearances in Westchester County, said Justin Pruyne, the deputy commissioner of Westchester County Department of Corrections.
Two police officers from the county, one of whom was driving, were also on the bus, authorities said.
Three people were rushed to Elmhurst Hospital with minor injuries, authorities added. It was unclear if the prisoners were among those hurt, and it was unknown if any criminality was suspected.