Durvesh Singh (20) and the girl were badly beaten by sticks by the girl's family in a cane field, killing the youth on the spot, they said.
They tried to kill the girl by strangulating her but as locals intervened, she was saved and later rushed to a hospital where her condition is stated to be serious, they said.
Senior Superintendent of Police (SSP) Santosh Kumar Singh said that it was a case of honour killing and all the accused would be arrested soon.