The decisive blows were landed by Nedim Bajrami, netting his first goal in Rangers colours inside less than a minute, and Ross McCausland who sprung from the substitutes bench in the second half to put the icing on the cake.
It was a display full of swagger from Philippe Clement’s side who, in the process of recording their third consecutive victory, registered their third clean sheet in a row.
Full focus will now shift to Sunday afternoon’s Scottish Premiership encounter at home to Hibernian.
Clement handed a first Rangers start to Neraysho Kasanwirjo, with Mohamed Diomande also replacing Dujon Sterling in midfield.
And it took less than a minute for the Light Blues to crack open the contest, with a familiar source of early strikes this year handing his side an early advantage.
Indeed, Dessers showed tremendous opportunism to latch onto a loose pass back towards the Malmo goalkeeper, rounding Johan Dahlin in the process and, when his finish from an acute angle came back off the post, Nedim Bajrami was perfectly placed to tap home his first goal in Rangers colours.
The away side were on the front foot early and had certainly spooked their hosts with their early endeavour.
Malmo took a while to establish a modicum of control in the contest and they had the ball in the net on 17 minutes when Isaac Kiese Thelin volleyed home from a free-kick, but the offside flag curbed any celebrations from the hosts.
Rangers’ attacking trio of Vaclav Cerny, Dessers and Bajrami were looking lively, with the latter shooting straight down the throat of Dahlin from distance after jinking his way inside from the left-hand side.
The game settled into a period of inactivity either side of the half-hour mark, although Rangers very nearly exploded back into action when Cerny’s deflected strike ricocheted into the vicinity of Diomande but the Ivorian could not bring the ball under control.
Opportunity then knocked for Cerny who was released beyond the Malmo backline by Bajrami following neat build-up play from Rangers, but the Czech unfortunately could not provide the finish with only the goalkeeper to beat.
Dessers and Bajrami combined soon after when the Nigerian drove menacingly into the Malmo defensive third, but the Albanian’s return pass lacked the required purchase and the home side cleared to safety.
The Swedish hosts began the second period the stronger of the two sides, piecing together a sweeping move in which captain Anders Christiansen ballooned over from a Hugo Bolin pull-back.
Still, Rangers continued to show an attacking flavour in their approach play; Lawrence spooning a difficult opportunity over the crossbar from Cerny’s wide delivery.
The Light Blues were being asked questions defensively – albeit Jack Butland was relatively untroubled – and the defensive line was responding, chiefly through a succession of well-timed interventions from John Souttar and Robin Propper.
Clement turned to his substitutes bench and McCausland almost made an immediate impact when he ghosted in at the far post and nodded wide from Bajrami’s well-worked cross.
And the Academy graduate was rewarded for his endeavour when he fired the Gers into an unassailable lead with 14 minutes remaining.
Indeed, his powerful, driven strike from the edge of the penalty area was dripping with accuracy as it clipped the inside of the post and left Dahlin with little chance in the Malmo goal to seal an impressive victory.
MALMO: Dahlin, Stryger, Rosler, Jansson (Rosengren, 67), Busanello, Pena, Johnsen (Jorgensen, 84), Berg (Loukili, 84), Christiansen (Botheim, 58), Bolin (Ali, 67), Kiese Thelin
Subs: Friedrich, Rieks, Olsson, Gudjohnsen, Makolli, Zatterstrom, Skogmar
RANGERS: Butland, Tavernier, Souttar, Propper, Kasanwirjo, Barron, Diomande, Lawrence (Raskin, 79), Cerny (McCausland, 67), Bajrami (Sterling, 83), Dessers
Subs: Kelly, Munn, Dowell, Balogun, King, Fraser, McKinnon, Lovelace, Curtis