The hosts scored and conceded in 4/5 last games, while Besiktas scored and conceded in 5/7 last matches.
Lyon president Jean-Michel Aulas was subsequently seen calling for calm inside the stadium via a loudspeaker.
Lyon said their fans sought refuge on the pitch after they were pelted with objects and flares from the upper tiers of the stadium.
At 8.16pm, with no members of the public remaining on the playing surface, Lyon tweeted that a revised kick-off time would be announced "in several minutes".
It is the third incident at a Uefa competition this week, following Tuesday's bomb attack on Borussia Dortmund's team bus and Wednesday's clashes between Leicester City supporters and police in Madrid.
The fans exited the pitch only after police intervened.
Football was left scarred by another sickening night of violence as supporters fought in the stands and fans raced to the pitch for safety before Lyon's Europa League tie with Besiktas.
There is a return match scheduled for April 20 in Istanbul, which reportedly has a hostile reputation against visiting fans.
On next week's scheduled second leg he added: "I don't know what needs to be done, play behind closed doors or elsewhere but it will be too risky for us to play this match with the same fans that were here tonight".
"Disciplinary proceedings have been opened following the UEFA Europa League quarter-final match between Olympique Lyonnais and Beşiktaş JK (2-1), played on 13 April in France", the statement read.
The trouble eventually abated and the teams were able to start the game 45 minutes later than planned, with Lyon recovering to beat the Turkish champions 2-1 in the quarter-final's first leg.
Football hooligans around the world are fighting in the stands, brawling in the streets and generally causing mayhem wherever they go.
Stewards looked to be injured from the violence as police and medics tried to safely take them away from the violence.