The Croatian star netted what proved to be the only goal of a dramatic game with a sensational overhead kick on 21 minutes which keeps the heat on league leaders Celtic.
It was a truly sensational effort from the in-form striker who put in a great shift for his side before being replaced by David Healy late on.
Jamie Ness returned to the Gers starting line up as Walter Smith was forced in to one change by the loss of Richard Foster.
The on-loan Dons defender was unable to face his parent club so American Mo Edu slotted in admirably at right back with Ness taking the vacant midfield spot alongside Steven Davis.
Steven Whittaker continued in the centre of defence, partnering captain Davie Weir, with Sasa Papac in his usual left back role.
Jelavic and Kyle Lafferty continued up front with service from the wings provided by Gregg Wylde on the left and Steven Naismith on the right.
Not for the first time this season, Smith was only able to name six substitutes prior to the kick-off.
The Rangers fans crammed in to the away section of the ground were in great form as both teams started this game at a frantic pace.
Despite a poor playing surface the early stages were open and the tactics from Craig Brown were clearly to get the ball to in to the Dons’ wide men Sone Aluko and Josh Magennis.
The latter was looking menacing and in the first ten minutes he produced two dangerous, low crosses.
The first was missed by Aluko but the same player connected with the second delivery and the ball curled away from the top corner of Allan McGregor’s goal.
At the other end a long range effort from Steven Naismith finished up in the Richard Donald Stand.
On 14 minutes positive play from Sasa Papac resulted in the Bosnian’s powerful, near post shot being palmed away nervously by Jamie Langfield.
It was end to end stuff and on 21 minutes on the clock it was Jelavic that opened the scoring and what a goal it was.
The ball dropped from the skies in to the 18-yard box and Jelavic, who was holding off Zander Diamond, sent a stunning overhead kick past Langfield to break the deadlock.
It was the Croatian’s 14th goal for Rangers and the best of his Ibrox career so far.
On 25 minutes referee Calum Murray produced his first yellow card and it was Jamie Ness that was the recipient having fouled both Aluko and Chris Maguirre.
But with Rangers a goal up the game finally settled down with the visitors comfortable in possession and the Dons trying their best to get back in to the game on a tough surface.
Clear cut chances were few and far up to the half-time whistle with Aluko and Robert Milson trying their luck at the death with ambitious long-range efforts which failed to trouble McGregor.
Rangers again started the second period positively and almost straight away a poor challenge from Derek Young on Naismith gave Murray no option but to award a free-kick 25 yards from Langfield’s goal.
The Croatian set his sights but the ball clattered off the wall.
From there Aberdeen regained their composure and Magennis again sent in a terrific cross to the back post which Edu did well to clear under pressure.
On 54 minutes an uncharacteristic mistake from Whittaker allowed Maguire and Vernon to counter with only Davie Weir in front of them.
Thankfully Edu used all his experience and pace to get back and his last ditch challenge meant that Maguire curled his effort around the wrong side of McGregor’s left hand post.
Prior to the hour mark Brown made his first tactical switch with his side in the ascendency with Clark Robertson replacing former Ibrox left-back Steven Smith.
And it was not long before Walter Smith did the same with El Hadji Diouf going on for Gregg Wylde before Kyle Hutton was thrown on for Ness.
With 65 minutes played it was Aberdeen that were threatening with Maguire causing all sorts of problems with his runs and tenacity.
At the other end Jelavic again tried his luck with another overhead effort but this time it was wide of target.
It was proving to be a pulsating night of football and on 72 minutes a clever training ground free-kick from 20 yards had the Aberdeen fans out of their seats.
Maguire surprised the Gers wall by feeding the ball down the left channel to Milstrom as opposed to shooting.
The on loan Fulham playmaker drilled the ball across goal and Edu again was there to clear off the line with the majority of the Red Army thinking the ball had gone in.
With time running out and the home side pushing men forward the game was opening up and Kyle Hutton tried to take advantage when he skipped past a challenge in midfield and unleashed a stunning 35 yard rocket which flew past Langfield’s right post with the keeper flapping at thin air.
In the end Rangers worked their socks off to retain possession with the full time whistle looming and they secured the three points their desire and commitment deserved.
They now move on to 74 points and can look forward to playing at Ibrox again on Saturday when St Mirren make the short journey along the M8 to take on the Champions.