Salihi volleyed home Armend Dallku's 37th minute cross as Josip Kuze's team followed a goalless draw in Romania with three points as Luxembourg slumped to a second straight defeat.
The visitors, who were thrashed 3-0 at home by Bosnia in their opener, finished with 10 men after Mario Mutsch was dismissed just before the hour for a second yellow card.
Albania, who now have four points along with Belarus, host Bosnia on Oct. 8 in their next qualifying match of the campaign.
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