Players from both sides were involved in a melee behind the Lyon goal after Everton defender Ashley Williams shoved opposition goalkeeper Anthony Lopes to the floor in the 64th minute.
Video footage showed an Everton fan holding a child appearing to slap Lopes on the forehead, while some other supporters alongside the advertising hoardings grabbed at Lyon players.
Williams equalised shortly afterwards but Bertrand Traore netted a 75th minute winner against Ronald Koeman's struggling side.
A statement from Everton said: "Club officials have reviewed the footage of an incident that occurred in the 64th minute of the game against Olympique Lyonnais where supporters towards the front of the lower Gwladys Street End became involved in a confrontation between players.
"We have identified one individual whom action will now be taken against. This will include a ban from attending future Everton fixtures and the club registering a formal complaint with Merseyside Police."
Merseyside Police will also review footage and work to establish whether criminal offences have been committed and made an appeal for witnesses to come forward with information.
"Merseyside Police can confirm officers are investigating following a disturbance on the pitch during a football match between Everton and Lyon at Goodison Park last night, Thursday October 19," a statement read.
"Detectives will be reviewing footage and speaking to witnesses to establish what, if any, offences have been committed.
"Anyone with information which could assist the investigation is asked to contact Merseyside Police on 0151 777 4065 or Crimestoppers anonymously on 0800 555 111."
After the match, Lopes reported he was hit by a fan but sought to play down the incident.
"It was no big deal, it happened and that was it. For me, it seemed to wake up the fans and they seemed to love it," he told reporters.
"I don't think it is part of the English atmosphere to hit an away player, but it was like this. But I am proud that we were strong at 1-1 we won the game despite everything. It was a very proud night for us."
Williams was led away from the fracas by Everton goalkeeping coach Patrick Lodewijks before he and Traore were booked.
"It's what happens, it's football. It is what it is," the Wales international told reporters.
"It's high emotions. We want to win the game, they want to win the game and stuff happens. That was just one of those times."