SPL - Jelavic bags brace in Gers victory
The striker netted the opener in the first half and doubled the lead from the penalty spot after the break.
Jon Daly got a goal back for the visitors, but United defender Garry Kenneth was then responsible for Rangers' third with an own-goal, with the only dampener on the day for the Scottish champions being the dismissal of Gregg Wylde late on.
Rangers made two changes from last week's 2-1 win at Aberdeen for the meeting with United - the last SPL team to beat the Light Blues back in April.
Kyle Bartley made his first appearance since August after recovering from a hernia operation and Matt McKay also started, with Steven Naismith out for the season with cruciate ligament damage and Lee McCulloch sidelined with a calf strain.
Kyle Lafferty was also absent from the squad, suggesting he had picked up a knock before the clash at Ibrox.
United were boosted by the return of Kenneth, after he missed last week's 4-1 win over Dunfermline, while Gary Mackay-Steven also started for the visitors.
The Tannadice side carved out a decent chance early on when Johnny Russell teed up the shot for Daly and he scooped just over after appearing to lose his footing and following some poor defending from Rangers.
United were threatening again when Scott Robertson whipped into the path of Russell and he had a go from the edge of the box but the effort fell well wide of target.
Instead it was Rangers who surged into the lead with 19 minutes gone. McKay delivered the cross from the right for Jelavic and he left goalkeeper Dusan Pernis with no chance as he bulleted a header into the back of the net.
Jelavic could have added to his tally when he met a Lee Wallace ball from the left with a volley in front of goal that Pernis did well to hold on this occasion.
That proved to be Wallace's last real involvement in the game as he was replaced by Wylde after 32 minutes after being forced out of the action through injury.
United could have hit back just before the break through Daly but goalkeeper Allan McGregor was able to smother at the striker's feet before he could connect properly with Paul Dixon's cross at the back post.
United emerged for the second half determined to haul themselves back into the game and Kenneth tried his luck with a speculative shot from distance that failed to trouble McGregor.
But their hopes were dealt a blow when referee Brian Winter pointed to the penalty spot after Wylde was felled in the box by Dixon.
Jelavic stepped up for the spot-kick and sent Pernis the wrong way with a composed finish to grab his second of the day with 63 minutes gone.
Dundee United threw themselves a life-line after 73 minutes when Daly met Kenneth's cross and looped a header past the grasp of McGregor.
But there appeared to be no way back for United when Steven Whittaker delivered a teasing ball into the box and Jelavic produced a diving header that was inadvertently directed into the back of the net by Kenneth with eight minutes to go.
Wylde was then shown a red card in the final minute for a challenge on Willo Flood as Rangers ended the game with 10 men but with three points.