Didier Deschamps' side were dominant for much of the game against Slovakian minnows but had only the defender's second-half header to show for their efforts.
And the visitors could easily have snatched a point, with Steve Mandanda making two fine saves to preserve his side's advantage.
Each team will feel they were harshly denied a penalty in the first half, with Zilina's Momodou Ceesay the first to be deprived when he was clearly floored by Diawara.
The Gambia striker's frustrations may have been increased when he was awarded a meaningless free-kick in midfield two minutes later after going down under minimal contact from Stephane Mbia.
In between times Andre-Pierre Gignac forced a good low save from Zilina keeper Martin Dubravka, who then kept out a similar effort from Andre Ayew.
Summer signing Gignac was denied by an even better stop from Dubravka in the 13th minute, the keeper diving full length and showing a strong left hand to parry away a well-struck drive.
The Marseille pressure continued and Cesar Azpilicueta, Lucho Gonzalez and Mathieu Valbuena combined fluidly down the right but from the latter's cross, Ayew sliced his volley tamely wide.
And the hosts had their own penalty shout on the half-hour as Gignac's effort was well blocked by Lubomir Guldan and Gabriel Heinze's follow-up struck the outstretched arm of Jozef Piacek.
A trademark Taye Taiwo strike hit Stanislav Angelovic and rebounded almost to the centre circle before midfielder Stefan Zosak had Zilina's one shot of the opening period. His 30-yard effort, though, was always curling over.
Marseille almost broke the deadlock on the stroke of half-time when Brandao glanced Lucho's corner across the face of goal, Gignac nodded back across and a falling Brandao hooked over from less than a yard under pressure from Ceesay.
Valbuena was harshly booked for simulation after being taken down by Angelovic's fair ball-first challenge in the penalty area but his team took the lead from the resulting corner - wrongly awarded after the ball went out off Gignac.
Lucho delivered the corner and Diawara's header was fumbled into the net by the hitherto impressive Dubravka. Bizarrely, Taiwo was booked for pretending to shoot the goalscorer with the corner flag during the celebration.
Ceesay was twice off target as Zilina sought an equaliser and Mandanda then saved bravely from Tomas Majtan.
The France stopper had to be at his best once more to tip away Robert Jez's left-foot drive, though he got away with dropping the resulting corner.
Heinze blocked Jez's shot as goalmouth action was at a premium thereafter, though the UEFA match delegate's eye may have been caught by an incident in which Zilina's Vladimir Leitner appeared to be targeted with a laser pen while taking a corner.
Zilina substitute Tomas Oravec flashed a last-minute effort over the bar but Marseille held on.