The 22-year-old attacking midfielder, who was ordered to pay Trabzonspor 100,0000 euros ($1.08 million), will also miss his team's Champions League round of 16 matches against Atletico Madrid later this month.
Trabzonspor lodged a claim with world soccer's governing body FIFA in April 2013 alleging that Calhanoglu had breached the terms of his contract without just cause when he signed for German team Karlsruhe in 2011.
Turkey international Calhanoglu was a minor at the time.
FIFA had ruled in favour of the Turkish club and imposed the ban that was suspended pending an appeal at CAS.
"We regret this decision which we cannot comprehend," said Leverkusen sports director Rudi Voeller. Leverkusen is not involved in the dispute with Calhanoglu having joined in 2014 from Hamburg SV.
"This is a big blow both for Hakan and for us. Although Bayer Leverkusen had nothing to do with this it is also punished. Now we will miss an important player during a key part of the season."
Leverkusen are ninth in the Bundesliga on 24 points.
($1 = 0.9260 euros)
(Reporting by Karolos Grohmann; Editing by Toby Davis)
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