Everton went ahead in the 17th minute when Theo Walcott played in Bernard and he skillfully jinked through the West Ham defence before sliding the ball under goalkeeper Roberto Jimenez at his near post.
Richarlison almost doubled the hosts' lead but his 22nd-minute effort came back off the post.
The Toffees were again denied by the woodwork as Walcott struck the crossbar with a ferocious hit from the edge of the penalty area.
Substitute Sigurdsson wrapped up the win with a magnificent finish of his own, curling into the top-left corner in the second minute of stoppage time.
Everton's third win of the campaign lifts them up to 13th with 10 points, while West Ham drop to 10th, two points better off.
Everton: 26 Oct v Brighton (A)
West Ham: 26 Oct v Sheff Utd (H)