City goalkeeper Tom Glover was left bloodied after being struck in the face by a metal bucket as fans from the Victory end stormed onto the Melbourne Rectangular Stadium pitch on Saturday.
Referee Alex King also suffered a cut to his head and was engulfed by fans at the 20-minute mark as City led 1-0, while a camera operator was struck in the head by a flare earlier in the match.
FA boss James Johnson said it had started a "show cause" process - the governing body's jargon for an investigation - into Melbourne Victory over the fan invasion and was working to identify the individuals involved.
"We will be moving swiftly and will be taking the strongest sanctions that are available," Johnson told reporters on Sunday, saying the pitch invaders were not football fans.
"This is an element that goes beyond football. It's an element that infiltrates our game and that really try to ruin it for the two million people who love our sport.
"And it’s those people that we will be targeting in this investigation and who we will weed out of the sport."
Johnson said he was happy to report that Glover and King were in good health, though shaken by the incident.
City said Glover had suffered a concussion and "severe lacerations" to his face but would be able to return to training once he had completed concussion protocols.
The pitch invasion shocked Australian football, with Socceroos goalkeeper Danny Vukovic, who plays for A-League side Central Coast Mariners, describing it as the "darkest day for football in Australia".
"Our game is in tatters," he wrote on Twitter late on Saturday.
"An absolute disgrace what happened tonight."
Melbourne Victory managing director Caroline Carnegie said the club would assist a police investigation into the incident and apologised to Glover, King and the camera operator.
"Victory is bigger than a group of fans that did the wrong thing," she told reporters on Sunday.
"There’s no place at our club, no place in the game, no place in the league for that sort of behaviour."
The pitch invasion occurred amid a fan backlash against the A-League's decision to sell its men's and women's title-deciding Grand Final matches to Sydney until 2025.
Supporters groups from both Melbourne teams had planned a walk-out from the stadium at the 20-minute mark to protest the move.
