The Toffees went down 2-1 to the French side at Goodison Park, a result which puts manager Ronald Koeman's job in further doubt as their poor start to the season continues.
Minutes before Williams scored to cancel out Nabil Fekir's early penalty, the Everton centre-back sparked the game's controversial moment when he shoved the Lyon goalkeeper as they challenged for the ball.
Several players got involved and a series of shoves broke out on the edge of the pitch while some players tried to break it up.
There was an unusual hand involved, however, as an Everton supporter carrying his child started pushing some Lyon players.
The French stars seemed in shock as they turned around and saw the child in his arms.
Williams and Bertrand Traore were the only ones to be cautioned in the wake of the chaos, and both went on to score within the next 10 minutes.
The result propels Lyon to second in the group on five points, while Everton stay stuck to the bottom with one from three games.