The visitors were looking to build on the win over Chelsea and they got off to the best possible start when Kurt Zouma rose highest to head home from a corner after just four minutes.
West Ham really struggled from the onset, but they had Lukasz Fabianski to thank for keeping the score down as he produced a couple fine saves to keep out Dominic Calvert-Lewin and Lucas Digne inside the opening 20 minutes.
The lead was inevitability doubled on 33 minutes as Benard slotted home from six-yards out after a well-worked passing move for his first league goal for the club.
The Hammers enjoyed more of the ball in the latter stages of the half, although they were unable to carve out any serious opening in the final third.
West Ham coach Manuel Pellegrini opted for a double change at the start of the second half as Javier Hernandez and Michail Antonio were introduced.
The double change seemed to have no effect on the Hammers' approach as they failed to carve out a single chance in the second half, much to the displeasure of the home support.
The Toffees nearly bagged a third goal in the final 10 minutes of the half, but Richarlison could steer his header onto the crossbar with only Fabianski to beat.
Nevertheless, 2-0 was how the score ended as the Toffees move up to seventh spot with their first victory in London for over two years.