Law enforcement sources tell TMZ, Landau Eugene Murphy Jr. -- who won season 6 of 'AGT' -- was arrested Monday night in Whitman, West Virginia for misdemeanor domestic battery and domestic assault.

We're told they got into it when he told her he was leaving her ... which escalated into an argument about who would care for their 6-month-old baby. That's when Landau allegedly started slapping her and then struck her under her left eye with a golf club. The brawl spilled outside, where he allegedly slammed her head on the sidewalk.

She was treated at a hospital, and cops later arrested Landau. He was released the same night on a $3,500 bond.

As for Landau's side ... his manager says the fight was sparked when the singer came home and found his home trashed -- he claims by the girlfriend. He says she destroyed the house with a golf club ... then attacked him with the club and a knife. He claims he was only defending himself, and apologized for the incident.