Visiting Spurs controlled large parts of the possession during the early exchanges, but they were unable to create any real goalscoring chance of note.
Their best chance fell to Jermain Defoe in the 17th minute, but he delayed his effort, allowing Everton defender John Heitinga to retreat and clear the danger.
Friedel had better luck moments later when he parried Royston Drenthe's deflected effort to safety.
As the half drew to a close, Jelavic went close to adding his second of the match, but Friedel got across his line to deny the 26-year-old's effort from long range.
David Moyes's Everton will move into the top half of the Premier League table if they hold on to secure the three points after the restart.