The big money January signing slotted confidently in his first start for the Toffees to give his new team its fourth consecutive victory at Goodison Park
Jelavic's 20th minute goal set Everton on its way to a victory that condemned Tottenham to its third consecutive Premier League defeat under Harry Redknapp as concerns over the team's top four credentials continued to grow.
The match started at a timid pace and it wasn't until the 15th minute that a genuine chance was created. Jermain Defoe broke the Everton offside trap and collected a through ball with John Heitinga in pursuit.
After getting into the box, Defoe looked to fire a left-footed shot on goal but was denied by the recovering Dutchman's lunge. An aggrieved Defoe looked for a penalty but referee Mark Halsey deemed the defender's tackle legal.
This sparked the home team into life and Marouane Fellaini's left-footed volley from a wide angle was saved by the outstretched leg of Brad Friedel minutes later.
As Tottenham struggled to deal with the movement of the midfield five Moyes deployed, Leighton Baines found Leon Osman running from deep on the left-side of the box.
Osman latched onto the pass and nipped in and past an overly aggressive Younes Kaboul before pulling the ball back for striker Jelavic.
Everton's January signing showed impressive composure to side-foot the pass beyond Brad Friedel from the edge of the box and mark his first start with a first goal.
Redknapp opted to revert Bale to his natural left wing role after starting from the right flank and Spurs began to see an improvement but chances created were slim and Everton was content to deny the visitors the space to find rhythm in possession.
Moyes would have been content at the break with an encouraging first-half lead whereas the contrast in the Spurs dressing room would have been as obvious due to their lack of incisiveness out on the pitch.
Spurs opted to stick with the same starting 11 as they emerged for the second half and they showed renewed vigor as Defoe found space on the left of the box for Scott Parker's pass.
His jink inside and right-footed shot tested Howard but the American's save was routine as he beat the ball away for a corner.
Bale's cross-field ball to Benoit Assou-Ekotto set Spurs in motion again and the French full-back's volleyed pass found Luka Modric just inside the box. The Croatian got slightly too under his snapshot though and failed to trouble Howard.
Everton, in truth mustered little of note by the 65th minute and only the introduction of Jack Rodwell stemmed the flow of Spurs attacks.
Rafael Van der Vaart replaced Sandro with 20 minutes to go but immediately Spurs had to deal with a rare Everton attack. Leon Osman was played in on goal on the right of the box but his drive was too central and saved by Friedel.
Five minutes after Van der Vaart's introduction, Spurs thought they were level. Assou-Ekotto's cross found its way to the back-post with Bale arriving.
As the ball jumped up and struck Bale on the chest, it fell to Defoe who turned it past Howard. Unfortunately for the striker he had wandered a yard offside making the decision to disallow the strike correct.
For all Tottenham's second-half possession, Everton has been bred by Moyes to hold on to leads such as this. Adding Phil Jagielka to an already sturdy back-line shored the home side's defense further still.
The fortune was with Everton and Spurs departed Merseyside with another damaging loss.
The Toffees have found their stride as they so often do in the second period of the season and move into ninth place but for Spurs, they will be looking even more anxiously over their shoulder as Arsenal and Chelsea continue to gather points. Arsenal can move to within a point of Tottenham with a win against Newcastle Monday.