The start of the quarter-final first leg game was delayed for 45 minutes after spectators spilled onto the pitch to avoid missiles hurled from the upper stands at the Parc OL.
Two fans were taken to hospital after being hit, said a police spokesman.
Police estimated between 15,000 and 20,000 Turkish fans from throughout France were milling around the stadium complex before the game. A group of Turkish fans from Germany smashed glass doors at the Lyon club megastore after they were unable to buy tickets for the match.
Turkish pro-government media were blaming French fans for the trouble. "The Lyon hooligans attacked children," said the Yeni Safal daily newspaper, with a picture it said showed a French fan waving a crutch in the direction of a child.
Sabah, another pro-government daily website, claimed: "Lyon hooligans attacked the Turkish fans, including women and children and afterwards the Besiktas fans responded."
Corentin Tolisso and Jeremy Morel scored in the last seven minutes as the hosts came from behind to win 2-1 ahead of next Thursday's second leg in Istanbul.
Lyon midfielder Lucas Tousart said: "It was difficult having to wait for so long before the start of the game not really knowing when we were going to go out.
"It was hard remaining focused knowing what was going on in the stands. The team lacked concentration in the first half and we conceded the goal. But after the break we were better and that helped us to come back, get the equaliser and also the winner.
Tousart added: "I've seen videos of what it is like in Besiktas. We'll have to try and disregard all that and concentrate on building on this victory."