Last Thursday's match at Parc Olympique Lyonnais was delayed by 45 minutes due to fans from the lower tier entering the field after objects were thrown at them from within the group of travelling supporters in the tier above.
Besiktas fans also broke through segregation and fought with home fans.
UEFA charged Lyon in relation to crowd disturbances, setting off fireworks, stairways being blocked, insufficient organisation and field invasion by supporters after Jeremy Morel capitalised on an error from Besiktas goalkeeper Fabri to snatch a 2-1 win.
Besiktas were similarly punished for crowd disturbances and setting off fireworks, along with throwing objects.
Under punishments handed down by UEFA's Control, Ethics and Disciplinary Body (CEDB), each club has been fined €100,000.
A statement released by UEFA read: "The CEDB has decided to exclude Olympique Lyonnais [and Besiktas] from participating in the next UEFA club competition for which it will qualify. This exclusion is deferred for a probationary period of two (2) years."
The statement added: "Based on the official reports and the current available evidence at the disposal of the UEFA disciplinary bodies, the CEDB has considered that the supporters of both clubs were responsible for the crowd disturbances at the match."
The second leg of the quarter-final tie takes place in Istanbul on Thursday and comes after Lyon were involved in another ugly flashpoint over the weekend.
In Ligue 1, their game at Bastia was initially delayed by home fans entering the field to confront Lyon players during the warms-ups, while a similar fracas at half-time resulted in the match being abandoned.