The Dons went close through Sone Aluko and Josh Magennis before the Croatian struck with a sublime finish in the 22nd minute.
The home side created a number of chances in the second half but were unable to find away past Rangers goalkeeper Allan McGregor as they slumped to a fourth successive defeat.
Gers youngster Jamie Ness was handed a start in place of Richard Foster, who was ineligible to play under the terms of his loan agreement.
Walter Smith's side, who earlier this season came from two goals down to win 3-2 at Pittodrie, went into the match five points behind their Old Firm rivals at the top of the table.
Zander Diamond returned for Aberdeen, who have won five of their previous nine meetings between the teams at their own stadium - but have been struggling for form in recent weeks.
The defender recovered from a stomach injury to replace Nikola Vujadinovic who started on the bench, while strikers Scott Vernon and Magennis were also included at the expense of Nick Blackman and Peter Pawlett.
The home side should have opened the scoring with less than three minutes on the clock when Magennis drove a low cross into the box for Aluko, who looked set to apply the finishing touch but failed to connect with the ball.
The Dons continued to search for the opener with McGregor making a smart save to claw away a Magennis volley.
At the other end Sasa Papac pounced on a careless pass by Ryan Jack to test Jamie Langfield in the Aberdeen goal.
Jelavic fired a shot from distance inches wide of the target - before was on target in spectacular fashion in the 22nd minute.
Kyle Lafferty played the ball into the path of Jelavic - and his overhead kick gave Langfield absolutely no chance.
Midfielder Ness picked up the first booking of the evening for a poor tackle on Chris Maguire as the match became an increasingly ill-tempered affair.
Undeterred by going a goal behind, Aberdeen looked to restore parity before the break with Aluko heading a shot just over the bar, but the visitors held on to take a one-goal lead into the break.
Maguire almost grabbed an equaliser for the Dons 10 minutes after the restart when he robbed Steven Whittaker of possession before curling an effort wide of the Rangers goal.
The Dons were then forced to make a change after 58 minutes when Steven Smith limped off, to be replaced by Clark Robertson.
Maurice Edu was cautioned for a late challenge on Maguire before Rangers introduced El-Hadji Diouf and Kyle Hutton for Gregg Wylde and Ness.
The Dons went close again with a well-worked free-kick which saw Maguire and Rob Milsom combine to set up Vernon - but his effort was blocked on the line by Lafferty.
With less than 10 minutes remaining, Blackman replaced Magennis as Aberdeen searched for the equaliser.
However it was the visitors who went closest in the dying stages when Edu's header was tipped away to safety by Langfield.