The Turkish club were handed the ban by governing body UEFA because of ongoing financial problems.
Besiktas were initially banned from the competition in April with UEFA's Appeals Body reaffirming the decision the following month.
That prompted Besiktas to take their case to the CAS, but that appeal has also been thrown out with the Turks excluded from the next two UEFA club competitions they qualify for, with the second exclusion suspended.
They must also pay a fine of 200,000 euros, of which 100,000 euros is also suspended.
Besiktas qualified for the Europa League after finishing fourth in the Super Lig last season, but details of the club's financial problems later became apparent, including non-payment of salaries and taxes.
Their Europa League place has been handed to Eskisehirspor, who face St Johnstone in the second qualifying round.