Joao Mario put West Ham ahead with his first goal for the club from 20 yards after Cheikhou Kouyate's left-wing cross.
Arnautovic quickly doubled the lead on the rebound after Alex McCarthy had saved his initial header.
The Austrian made it 3-0 on the stroke of half-time when he swept home from Arthur Masuaku's cross.
It gave West Ham three first-half goals in a Premier League match for the first time since September 2012.
Southampton offered little in Mark Hughes's first league match in charge and stay in the bottom three.
The Hammers ended a three-match losing run to move five points clear.
West Ham: 8 Apr v Chelsea (A)
Southampton: 8 Apr v Arsenal (A)