Malmo went straight on the offensive from the opening whistle and although nothing came from two corners, it told Chelsea that they would be tested by a side who suffered just one defeat during the group stages.
However, that is arguably what an out-of-form Chelsea required and the Blues controlled the opening half-hour of the contest without creating a single chance until they eventually went ahead through Barkley.
Pedro's cross from the right should have been dealt with by a defender but the delivery went through to Barkley, who had to get the ball out of his feet before poking a shot into the bottom corner from eight yards out.
Going behind led to Malmo going onto the front foot with their relentless pressing forcing Chelsea into a number of mistakes in their own final third, and the home side went close to an equaliser when Christiansen's long-ranger was deflected marginally over the bar.
Chances were few and far between during the opening 45 minutes, but it was a different story at the start of the second half as Malmo pushed for a leveller.
The hosts appeared the more likely to score the second goal of the game, but Chelsea doubled their advantage shortly before the hour mark when good work from Willian gave Giroud the chance to cleverly flick the ball into the net from close range.
Chelsea soon went in search of a third and it should have arrived when Giroud rolled the ball into the path of Barkley, but the midfielder was superbly denied by Johan Dahlin from near the penalty spot.
However, Malmo found a way back into the contest with 10 minutes remaining. Markus Rosenberg threaded a pass through to Christiansen and the midfielder found the bottom corner of the net from 15 yards after keeping in front of Mateo Kovacic.
Chelsea should have reestablished their two-goal advantage just before the full-time whistle, but Giroud failed to hit the target at the near post after being found by Hazard's low pass.
The forward then fired too close to Dahlin from an acute angle after again being teed up by Hazard, but Chelsea and Sarri will be relatively satisfied with possessing a lead and two away goals ahead of the return game in West London.
MALMO (4-4-2): Dahlin; Vindheim, Nielsen, Bengtsson, Safari; Bachirou, Traustason (Lewicki 70'), Christiansen (Gall 82'), Rieks; Antonsson (Strandberg 70'), Rosenberg
CHELSEA (4-3-3): Arrizabalaga; Azpilicueta, Christensen, David Luiz, Palmieri; Jorginho (Kante 74'), Kovacic, Barkley; Willian (Hazard 71'), Giroud, Pedro (Hudson-Odoi 84')