Neither goalkeeper was called into any telling action before the break, with both teams finding it difficult to create clear-cut chances.
Everton's best chance fell to Steven Pienaar, who was unable to keep his rising effort on target.
Up the other end Stephane Sessegnon sent an overhead kick attempt wide of Tim Howard's goal as the half drew to a close.
David Moyes's side continued to dominate and extended their advantage further when Pienaar, who was unmarked on the edge of the area, converted a pass from Gueye 14 minutes from time.
The points were sealed just a minute later when Gueye claimed another assist, picking out Osman who beat Mignolet from long range via the aid of the post.