Southampton were ahead in the sixth minute when Adam Lallana corner evaded the Everton to defence to allow Ramirez to apply a simple finish at the back post.
Johnny Heitinga could have equalised just moments later but the Dutch defender fired his header wide from close range.
The visitors were encouraged by their good start and could have doubled their lead when Jay Rodriguez found himself with only Tim Howard to beat. However, the former Burnley forward failed to hit the target.
Everton ensured that they were level midway through the first period after Leon Osman pounced on a loose ball inside the box to drive in a powerful shot.
Jelavic was close to putting the hosts ahead a few minutes later but the Croatian striker's curling effort landed just wide of the post.
However, the clinical finisher made up for his uncharacteristic miss by collecting a pass from Kevin Mirallas and calmly finishing from inside the box.
Jelavic was at the centre of the majority of Everton's moves and the former Rangers forward added a third on the stroke of half-time with a simple header from close range.