On the night Rikard Norling’s side, who are 17 games in to the Swedish Allsvenskan season, looked sharp and well organised and the only goal came from Daniel Larsson, the club’s top scorer, on 18 minutes.
It was poor defending from the Light Blues initially but when the impressive Wilton Figueiredo flicked the ball in to his teammate’s path his finish from 18 yards was hugely impressive.
Rangers, of course, had chances to get back in to the game with Steven Naismith coming close on two occasions on the hour mark while Jelavic was missing the scoring touch that made him an instant hero last term.
Lee Wallace was given his first run-out in front of the Ibrox faithful, turning out in the Champions League five years after debuting in the competition for Hearts.
The 23-year-old was one of two changes to the team that faced Hearts on Saturday with captain Davie Weir returning to the side for kick off.
The club’s inspirational skipper and Wallace replaced Kirk Broadfoot and Juanma Ortiz with the latter starting in a wide left role, which was mirrored on the opposite flank by Steven Whittaker.
Steven Davis moved in to the middle of midfield, alongside Lee McCulloch and Mo Edu, leaving Naismith to partner Jelavic in attack.
The match also marked Jelavic’s first involvement in Europe for Rangers with the Croatian ineligible in continental competition last season.
Fans of both sides perfectly observed a minutes silence for the victims of Oslo and Utoya in Norway and after a familiar huddle from the Malmo players the action got underway with Ibrox Stadium bathed in beautiful sunshine.
But with just a minute on the clock Larsson showed his threat when he sprinted clear of the Gers backline but his prodded shot beyond McGregor lacked the power or accuracy to find the net and Papac was able to clear.
At the other end Jelavic felt that he had been wrestled to the ground by Malmo skipper Daniel Andersson but Spanish ref Antonio Lohez quickly told both players to get to their feet, refusing to point to the penalty spot.
Norling’s team looked confident in the early stages and on 8 minutes Brazilian midfielder Figueiredo skipped past Mo Edu and his powerful drive across goal was palmed away by McGregor.
The green-booted Jimmy Durmaz then skewed the rebound in to the Copland Road Stand from a wide angle when he should have hit the target.
Rangers, however, were tidy in possession and when Steven Whittaker cut inside and swerved a left-footed, inswinging cross Jelavic, who was being tightly marked, could not get enough purchase on the ball to trouble Czech shotstopper Dusan Melicharek with his header.
On 18 minutes the deadlock was broken and it was the away support that were celebrating a crucial away goal.
Whittaker was caught on the ball on the edge of his own box and when Figueiredo flicked the ball cleverly in to the path of Larsson he unleashed a stunning strike from 18 yards which rippled the roof of McGregor’s net at his near post.
It was a fantastic finish but it was also a gift for the Swedes who won the ball to easily when it should have been punted clear.
And before the hour mark the home side suffered another blow when the injured Weir had to be replaced by Ortiz.
This led to Papac moving in in to the centre of defence and Wallace taking up his usual left back role with Ortiz ahead of him.
As soon as the Spaniard was introduced Naismith won a free-kick twenty five yards out and this was reduced to twenty after Jelavic’s initial effort was stopped by a Malmo player in the wall with the use of a hand.
Steven Davis stepped up for the second attempt and his shot brought out an excellent stop from Melicharek who dived to his right to divert the ball around the post.
With five minutes of the half remaining Rangers came close to levelling the tie when Davis’ teasing delivery was glanced on by Naismith but the ball arrived quickly at Jelavic’s body and it appeared to hit off his hip before squirming inches wide of Melicharek’s goal.
This was to be the last chance of opening period for the Light Blues who still looked to be finding their feet against their lively opponents who were dangerous and clever on the counter attack.
This was again evident just 60 seconds after the restart when the pacey Figueiredo beat Papac to a long ball down the right wing and his cross found Larsson who, having evaded Bougherra with a clever dip of his shoulder, kicked at fresh air in a great position.
On 49 minutes Larsson again caused a problem when he used his electric speed to win a free-kick on the left flank and Figueiredo’s dipping set piece had to be turned over the bar impressively by McGregor.
At this stage Rangers needed a spark and Lee McCulloch tried to supply it when his daisy cutter from 25 yards brought out a good save from Melicharek.
McGregor, however, was soon back in the thick of things and before the hour mark he had to be at his athletic best to pull off a wonder stop from Figueiredo who tried his luck from 30 yards.
And from the resulting corner Scotland’s number one had to be focused to deny Malmo’s skipper Daniel Andersson who looked to find the corner of the net after rising to meet a header.
It was end to end stuff in the second half and on 66 minutes it was Rangers turn to attack.
Jelavic pulled the ball down brilliantly on the right and his great cross to the back post was narrowly missed by Naismith who slid in but could not send the ball towards goal on the stretch.
Just two minutes later and it was a case of deja vu when the two forwards combined in exactly the same way but on this occasion Naismith should have scored with the goal gaping.
It was, however, better from McCoist’s men and on 73 minutes Melicharek again performed heroics to make a point blank stop from Jelavic who turned quickly in the box after excellent play from Wallace.
On 78 minutes Rangers had another chance to level when Naismith teed up Davis who fired a shot over the bar before Jiloam Hamad and substitute Dardan Rexhelpi fired raking shots which had McGregor had to deal with.
And In the dying minutes a floated cross in to the box from Ortiz found Jelavic but, with just the keeper to beat, he failed to get his head on the ball and another great opportunity was missed.
This just about summed up Rangers night but having created numerous chances throughout this 90 minutes Ally McCoist and his players should feel confident that they can get the goals they require when they travel to Sweden next week.