The big Croatian looked hungry throughout and took his 34th minute goal brilliantly after Madjid Bougherra had scored his first league goal of the season.
After suffering a shock 1-0 defeat at Tynecastle on Saturday it was all about getting back to winning ways for the Light Blues and Smith made two changes to his line-up.
Jelavic was joined up front by Steven Naismith with Kyle Lafferty dropping out of the first XI while Vladimir Weiss also featured on the left flank in Mo Edu’s place.
Andy Webster missed the game altogether due to a hamstring injury.
It was Hibs that got the match underway and immediately they pushed up the park.
But a sloppy pass in midfield allowed Weiss to dart 60 yards down the left flank and after turning former-Celt Richie Towell inside out his cross had slightly too much power for Naismith who was lurking at the back post.
Just four minutes later and the attacking duo combined again with Weiss playing a delicious pass to Naismith in the 18-yard box but his first touch was poor and Graeme Smith gathered.
Going in to tonight’s game Colin Calderwood’s men had yet to score a goal in 2011 but Colin Nish did have the ball in the net when he lobbed Allan McGregor in front of the travelling Bears.
Thankfully, the Main Stand linesman had already raised his flag and more than a few sarcastic cheers followed.
With 15 minutes on the clock neither side had found their rhythm with the bobbly Easter Road surface not helping matters.
In fact the first shot on goal did not arrive until 22 minutes when Hibs no. 22 Danny Galbraith tried his luck from distance but his effort was straight at McGregor.
At the other end Kevin McBride became the first player to receive a yellow card for a desperate challenge on Weiss and from the resulting free kick the visitors took the lead.
Weiss floated the ball in to the box and when the ball dropped to Bougherra he lashed a brilliant right foot shot across Graeme Smith from a tight angle.
It was a finish Jelavic would have been proud of and straight away Rangers went in search of a second.
On 31 minutes Weiss easily dispossessed left-back Callum Booth and his early, teasing delivery was inches away from Jelavic who slid in to the six yard box but could not get to the ball.
But moments later the club’s summer signing found the net and he savoured the moment with his teammates.
An under-hit back pass by Towell was seized upon by the Gers no. 18 who showed his class by rounding Smith and rolling the ball home with his cultured left boot.
With their side two goals down the Hibs fans voiced their frustration at the half-time whistle knowing only too well that the second half would be an uphill battle.
They did, however, come out fighting and McGregor was fortunate that a power packed shot from Murray flew straight in to his arms.
A low free-kick from Merouane Zemmama then crept under the Rangers wall and the club’s no. 1 had to be sharp to make a good stop at his right-hand post.
But with Hibs now under pressure to get a goal gaps were appearing all over the pitch and you felt it was only a matter of time before the likes of Naismith and Weiss exploited them.
On 63 minutes Calderwood made a double substitution with Derek Riordan and David Wotherspoon going on and it was not long before Mo Edu also replaced Weiss.
But with 19 minutes left there was a moment of real controversy involving Jelavic after he had tangled with defender Paul Hanlon who looked to leave his boot in as both players fell to the ground.
As they got to their feet Hibs keeper Smith ran to referee O’Reilly claiming the Gers hitman had headbutted his teammate.
From the stands the contact looked minimal but after an anxious wait the official decided to book both players as opposed to going for a red.
As a precaution Smith moved swiftly and replaced Jelavic with Kyle Lafferty but there was to be more concern for the manager when Papac took a sore one from Towell who caught the Bosnian with a late challenge.
Fortunately the left back returned and in the closing stages both Naismith and Steven Davis tested Smith from distance but Rangers could not add to their two first-half goals.
In the end it was a strong away performance from the Ibrox men who will now be looking to build on this victory in Sunday’s League Cup semi-final showdown against Motherwell at Hampden.