Glenn Murray scored what proved to be the game's only goal 25 minutes in - his sixth top-flight goal against West Ham in seven games - to earn Brighton their first win since August.
The Hammers registered 16 attempts as they chased a leveller in the remaining three quarters of the match, but a failure to find a way through brings an end to their recent upturn in fortunes.
A good block from Murray then prevented Arnautovic from picking out the bottom corner from a well-worked routine, and that would prove a key moment as the English striker edged his side in front up the other end soon after.
Murray fired into the corner from close range after getting on the end of Beram Kayal's delivery to maintain his impressive scoring run in this fixture.
Brighton, beaten just once in their last 17 league matches in the month of October, were in the ascendancy after that breakthrough goal and nearly had a second through Alireza Jahanbakhsh's drive.
Brighton had gone nine league matches without a clean sheet ahead of this one, and that run should have been extended when Fabian Balbuena was picked out at the back post, only to nod the ball wide.
Dunk's block took the sting out of Michail Antonio's shot to make it easy for Ryan to save, while up the other end Murray nodded just over the crossbar as the Seagulls looked to kill off their opponents.
United would squander one final opportunity in added time, with Arnautovic skying his shot when picked out by substitute Lucas Perez.
BRIGHTON & HOVE ALBION (4-4-1-1): Ryan; Bruno, Duffy, Dunk, Bong; Knockaert, Propper (Bissouma 80'), Kayal, March (Locadia 86'); Jahanbakhsh (Izquierdo 71'); Murray
WEST HAM UNITED (4-1-4-1): Fabianski; Zabaleta, Balbuena, Diop, Masuaku; Rice; Yarmolenko (Perez 70'), Noble (Snodgrass 83'), Obiang (Antonio 60'), Anderson; Arnautovic