Southampton had chances to extend their lead but were guilty of poor finishing, with Jay Rodriguez a main culprit, when he fired wide after being put through on goal against Howard.
The Saints were made to pay for their lack of quality inside the area as Leon Osman reacted well to a loose ball to smash in an equalsier for Everton.
The game's complexion changed in the space of six minutes before half-time, with Jelavic netting twice, to the delight of David Moyes.
Kevin Mirallas found an eager Jelavic just inside the box, allowing the Croatian striker to easily finish with time to spare. The former Rangers forward added his second with a simple header from close range.
Southampton continued to look bright in the second period and Ramirez was inches away from scoring his second of the game after lobbing the ball over Howard, only to see his superb attempt roll just wide of the post.
Howard's job was far from done as the game entered the final minutes in Merseyside and the American earned his wage with a brilliant save to deny Rickie Lambert and secure the result for the Toffees.