It was no more than Everton's early play deserved, with Roberto Martinez's side controlling the opening 10 minutes of action.
Danny Ings had a half-chance for the home side after 11 minutes, but his low effort was wide of the post.
Romelu Lukaku was next to come close down the other end, but his effort was always moving away from goal.
Everton were back ahead just before the 30-minute mark, however, when Lukaku bundled home from inside the Burnley box.
Jutkiewicz came close to levelling for the home side less than five minutes later, but the striker fired wide of the post after breaking into the Everton area.
Burnley continued to press in the latter stages of the half, but they could not find a leveller, with the Toffees coming closest to adding to the first-half scores when Steven Naismith hit the crossbar with a header.