Skipper Vossen scored the only goal after 31 minutes as Thun spurned plenty of chances to take something from the game.
Genk had already ensured they would finish top of Group G before the trip to Switzerland and, with a trip to Jupiler Pro League leaders Standard Liege awaiting them on Sunday, Been made five changes to his starting XI.
It seemed to have an unsettling effect on his side as Thun enjoyed the best chances in the first half-hour.
Genk goalkeeper Kristof Van Hout tipped a Marco Schneuwly effort wide of his post on 15 minutes before Milos Krstic shot over when well placed.
Nelson Ferreira attempt to lob Van Hoult went over and moments later Thun were made to pay for not converting one of their chances.
That it was a clever Vossen finish that opened the scoring only served to underline their wastefulness. The Genk skipper poked past David Moser with the outside of his foot after Ayub Masika muscled his way down the left to deliver the telling cross.
Thun continued to pass up good chances after half-time and when Marco Schneuwly and Krstic failed to find the target, coach Urs Fischer responded with a double change on 59 minutes.
Krstic and Schneuwly's brother Christian made way with strikers Berat Sadik and Josef Martinez coming on. Three minutes later Sadik shot over from a Benjamin Luthi corner.
Martinez tried his luck from 30 yards and while Van Hoult was at least worked he comfortably held on, as did Genk, to give Been a night to remember his landmark birthday by.