The Turkish side, managed by Slaven Bilic, were initially banned by UEFA from participating after allegations of match-fixing in their 2011 Turkish Cup final win over Buyuksehir Belediyespor.
They were temporarily reinstated while their appeal was heard by the Court of Arbitration for Sport (CAS), but the Swiss-based body has now confirmed Besiktas will not compete in the second-tier continental competition this season.
A CAS statement read: "CAS has today dismissed the appeal filed by Besiktas JK against the decision taken by UEFA to exclude it from the European Europa League 2013-14.
"Due to the urgency of the matter, the CAS Panel has issued its decision without the grounds. The full, reasoned award will be issued as soon as possible."
Besiktas battled back from a 2-1 first-leg deficit in their play-off with Tromso on Thursday night, winning the return 2-0 to secure a 3-2 aggregate triumph.
However, it will be Tromso who advance after UEFA's emergency panel held a meeting on Friday morning at which it decided the Norwegian team should be included in the draw for the group stage.
Fenerbahce, who also had their ban upheld, will be replaced in the Europa League by Cyprus club APOEL after a mini-draw of sides who missed out.
Fenerbahce would have otherwise dropped into the secondary European competition following their Champions League play-off defeat to Arsenal.
Finally, AZ Alkmaar and Atromitos will conclude the final 33 minutes of their second leg on Friday morning after the original kick-off was abandoned after a fire broke out at the AFAS Stadion with the Eredivisie side 3-2 ahead on aggregate.