Murray stroked home Beram Kayal’s excellent cross from the left on 25 minutes for the only goal of the game.
The Hammers responded positively, with Marko Arnautovic threatening to equalise before half-time.
The visitors continued to press in the second half, but Brighton held on to claim all three points.
Fabian Balbuena had the Hammers’ best chance, heading Felipe Anderson’s whipped corner wide from four yards out.
And Arnautovic lifted a late shot over the crossbar after good work down the left.
The hosts had opportunities to extend their lead, though, and Murray twice headed over the crossbar.
But his first-half goal was ultimately enough for the Seagulls to secure only their second win of the season.