Professional Footballers' Association chief executive Gordon Taylor has called the attack on Jack Grealish "disgraceful" and "cowardly" and believes the authorities must work with the government to help stamp out such incidents.
Aston Villa captain Grealish was punched by a fan during his side's derby match at Birmingham on Sunday, while later in the afternoon Manchester United defender Chris Smalling was also confronted on the pitch at Arsenal.
Those incidents came after a supporter confronted Rangers player James Tavernier during their game against Hibernian at Easter Road on Friday.
After a spate of incidents, Taylor insist the authorities must "put a lid on the problem" quickly and he has called on the Football Association to work with the government, police and stewards at the grounds to help stamp out such behaviour.
"It was disgraceful, cowardly and criminal. We have seen an increase in such behaviour at clubs this season. It's been racist abuse, it's been anti-Semitism and now it's real fear for the safety of players," Taylor told Sky Sports News.
"From that point of view, we had a meeting just the other week with the Minister for Sport about engaging on this issue between the police and the stewards, which is crucial, and the responsibility of clubs about trying to get the right balance between educating supporters but also having sanctions to make sure we put a lid on it because we've been down this road before and we don't want to go there again and see the game threatened."
Minister for Sport Mims Davies called the incidents "a disgrace" and added: "It is right that they are investigated immediately and strong action is taken by the football authorities and clubs.
"Protecting players' safety is of paramount importance and every possible step must be taken to ensure that they are not put at risk."
Birmingham fan Paul Mitchell, 27, pleaded guilty to assaulting Grealish at Birmingham Magistrates' Court on Monday.
Birmingham said Mitchell had earlier said would be banned for life from St Andrew's and Arsenal later pledged the same punishment would be imposed on the pitch invader at their match at the Emirates Stadium.
When asked about what sanctions should be implemented, Taylor said: "That's a job for the Football Association, but it's also a job for the police when the culprits are caught because these are criminal acts.
"If it happened on the high street then severe action would be taken. There's no reason why that shouldn't be done just because it's a football ground.
"There's a combination of sanctions you can consider - fines, ground closures, playing games behind closed doors and points deductions. Every case needs to be looked at on its merits but it has to be a job for the national association.
"I do believe it needs football getting together with government, as it has in the past, with the police and with stewards. I just feel it does need a coming together for us to deal with it."
Birmingham defender Harlee Dean told Press Association Sport after the match: "You should never fear for your safety on a football pitch and it's disappointing to see.
"There's a lot of hate amongst the fans and stuff, which is fine to an extent, but you shouldn't be able to come on the pitch and make contact with a player."
Former England striker Alan Shearer has warned players' lives could be at risk unless the authorities take urgent action to prevent pitch invaders and he wants the FA to come down hard on Birmingham to act as a deterrent.
"It is absolutely disgusting and if we don't stamp it out now, next time it could be a knife," Shearer said in his column in The Sun.
"It's that serious. Where are we at in football when some thug thinks that is what he is going to do? The football authorities have to be seen to nail Birmingham for the behaviour of their fan.
"If that means docking them points, playing in an empty stadium, so be it. There cannot be a strong enough punishment. It's completely ridiculous - I can't remember seeing anything as blatant as this.
"There has to be the right message here. Come down like a ton of bricks on Birmingham and punish so-called fans where it hurts, with bans and fines.
"It just proves we have a real problem. The authorities have to stamp it out now."
Former Villa midfielder Lee Hendrie backed Shearer's calls for urgent action, telling BBC Radio 5 Live: "It could have been someone running on with a knife - how are you meant to stop this?
"It's going to take something, that's going to cause a big, big scene in football, that's going to make people really stand up.
"Why do we need to wait for that to actually happen before someone gets really hurt? Imagine Jack's family sitting at the side and thinking about what could possibly have happened."
Meanwhile, United defender Smalling took to Twitter on Monday morning to reflect on what happened during his side's 2-0 defeat at Arsenal.
He said: "Commenting would only add to the attention he craved... Disappointed with yesterday's result but we'll learn from it. Looking ahead. Have a good start to the week everyone!"