Jelavic's impact had been limited to two substitute appearances due to abdominal problems since making a £6 million move from Rangers in January.
But the Croatia striker marked his first Premier League start with a first goal to extend Everton's unbeaten league run to seven matches and move them within two points of Liverpool ahead of Tuesday's Merseyside derby at Anfield.
For Spurs, it was a third straight Premier League defeat for the first time since manager Harry Redknapp, who has been linked with the vacant England job in recent weeks, replaced Juande Ramos in 2008.
Redknapp's side are only four points clear of fourth placed Arsenal and their local rivals can close that gap with a win over Newcastle on Monday.
Seamus Coleman started on Everton's left flank, due to on-loan Tottenham winger Steven Pienaar being ineligible, and the Irishman went close with just a couple of minutes gone, thumping wide from 20 yards.
Tottenham threatened for the first time when Gareth Bale took a throw quickly for Jermain Defoe to surge through, holding off Sylvain Distin only to slice a disappointing shot well wide of goal.
Everton carved out the first decent chance when Australia midfielder Tim Cahill laid off for Marouane Fellaini and the Belgian's curling shot was blocked by the legs of Brad Friedel.
But Friedel had no answer to Jelavic's 22nd minute strike as the Toffees took the lead in slick fashion.
Leighton Baines played through for Leon Osman, who surged into the area and pulled back for Jelavic to calmly shoot past the Tottenham goalkeeper.
Everton continued to press and, after cutting in off the right flank, Dutchman Royston Drenthe's shot cannoned off Ledley King and was held by Friedel.
But Bale was always dangerous and, after stepping in-field past Coleman, he thumped a drive wide from 30 yards and then narrowly missed with a free-kick from a similar distance.
As Everton struggled to clear, Luka Modric had a long-range effort held by Tim Howard, but Friedel was forced into a scrambling save to keep out Jelavic's powerful free-kick just before half-time.
Spurs wasted no time in pressing after the restart and, after yet more clever work from Bale, Defoe forced a flying save from Howard.
Howard once again denied Defoe after the striker was slipped through by Scott Parker.
Former Everton forward Louis Saha was brought off the bench 10 minutes into the second period as Redknapp looked to spark his team, but the next opportunity fell to Sandro, who headed wide at the far post after being left unmarked for a Modric corner.
Younes Kaboul also missed with a free-kick and Howard managed to grab another deflected Defoe effort as Everton struggled to get out of their own territory.
As Saha attempted to worry his old team, his prod at goal flew straight at Distin's face before Friedel did well to block Osman after a quick Everton breakaway.
Defoe had another effort correctly ruled out for offside before King headed wide from a lofted free-kick from Bale.
The hosts had to hold on for five minutes of added time, with Howard denying Kaboul's header from a Benoit Assou-Ekotto cross before Saha had a late effort turned on to the post by Howard.
Source: AFP
Source: AFP