The striker, back after ankle surgery, has been tipped to fill the void left by Kenny Miller's departure and he rose to the challenge at Easter Road, after Madjid Bougherra set the ball rolling with the opener in a 2-0 win.
The victory saw Rangers bounce back from defeat to Hearts at the weekend, while Hibs remain in the relegation zone after failing to repeat their impressive win over the Scottish champions at Ibrox the last time the two clubs met.
As well as Jelavic, Vladimir Weiss was also drafted into the side and Maurice Edu and Kyle Lafferty were dropped to the bench for Rangers.
Richie Towell made his debut for Hibs, after joining on loan from Celtic this week, and Colin Nish was also given the nod for the home side as Steven Thicot and Darryl Duffy were named among the substitutes.
The game was less than a minute old when Weiss embarked on a surging run from deep in his own half into the opposition box before delivering across the face of goal where Steven Naismith just failed to connect at the back post.
The pair combined again when Weiss squared from the left and Naismith bore down on goal only for goalkeeper Graeme Smith to race off his line and save well at the Rangers player's feet before he could unleash a clean shot.
Jelavic then attempted to turn provider for Naismith but he took a touch that was just too heavy and again allowed Smith to smother, before Allan McGregor gathered a tame shot from Danny Galbraith at the other end.
Rangers had looked the team most likely to find the back of the net and they opened the scoring with 26 minutes gone.
Weiss dusted himself off after winning a free-kick and lofted into the packed box where a headed clearance fell kindly for Bougherra to send an angled volley into the bottom corner.
The visitors could have doubled their advantage when Weiss whipped a tempting cross from the right into the path of Jelavic but he was just too slow to connect with the goal at his mercy.
However, the Croatian hitman was on the scoresheet after 35 minutes when he pounced on a poor back-pass from Towell before coolly rounding Smith and slotting into the empty net from six yards.
Hibs emerged for the second half determined to haul themselves back into the game and Ian Murray was not too far away with a ferocious volley from the edge of the area that was gratefully blocked by McGregor.
The Rangers goalkeeper was called into action again, this time to parry a curling Merouane Zemmama free-kick past his right-hand post after Jamie Ness had felled Galbraith 25 yards out.
Hibs had replaced Kevin McBride with Edwin de Graaf at the break and they also threw David Wotherspoon and Derek Riordan into the action with just over an hour gone. Edu then replaced Weiss for Rangers.
Jelavic was withdrawn for Lafferty seconds after picking up a booking for clashing with Paul Hanlon, who was also booked for his part in the altercation.
Rangers could have ended any hopes of a comeback when Naismith tried to pick out Lafferty as he bore down on goal but Smith intercepted and swiped to safety before he could stab home.
The big striker then turned provider with a cut-back for Steven Davis who saw his point-blank shot denied by a great save from the goalkeeper, as Rangers held on for the points anyway.