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The visitors had the first opening when Stephane Sessegnon was presented with a one-on-one opportunity thanks to a Stephen Fletcher through pass, but was unable to find a way past Tim Howard, who did well to keep it out.
Everton's main threat was coming from Kevin Mirallas in the early periods but the home side were unable to force Simon Mignolet into action.
The Belgian keeper was called upon just before the half-hour mark to keep out Steven Pienaar's lashed effort.
Moments later the home side were forced into a change as the energetic Mirallas had picked up a knock and was replaced by Steven Naismith.
Soon after, Everton were appealing for a penalty when Seamus Coleman was felled by James McClean but referee Lee Mason correctly awarded a free kick as the challenge was outside the area.
Jelavic had the chance to put the hosts ahead through the resulting set piece but he scuffed his volley from Leighton Baines's delivery wide.
Sunderland took the lead moments before the half-time whistle when Johnson half-volleyed Craig Gardner's centre past Howard to become only the second red-and-white player to score this season along with Steven Fletcher.