The Romanian minnows looked nervy from the offset, and were punished after 12 minutes when Caner Erkin nodded in Gokhan Gonul’s cross from close range.
However, Vaslui produced a shock equaliser three minutes later, as Marius Niculae stabbed home a cross from Lucian Sanmartean at the far post.
Ousmane N’Doye’s penalty was saved by Volkan Demirel before Fenerbahce finally surged ahead in the 71st minute, Kuyt ripping in a powerful effort into the top corner from another Gonul pass.
The Dutch striker was not done however, and put matters beyond any doubt in the 76th minute with a simple finish after Vaslui’s defence had switched off from a free-kick.
Moussa Sow put the icing on the cake late on with an unstoppable half-volley from just outside the area and Fenerbahce now await the draw for the final qualifying round before the Champions League group stages.