Tottenham striker Emmanuel Adebayor was subjected to the racist chanting on several occasions during his team's 4-1 defeat to Inter, while one home fan was also seen waving an inflatable banana.
Despite repeated attempts by UEFA and FIFA to stamp out racism in football, the problem keeps rearing its head and Bernstein admits the regularity of such abuse is worrying for the game.
"I condemn the racist abuse that Tottenham Hotspur players suffered at Thursday night's Europa League fixture in Milan," Bernstein said in a statement on the FA's website.
"It is unacceptable and concerning that clubs in Europe have had to endure recurrences of abuse this season.
"I am sure UEFA will take appropriate action but it also reminds us of the need to work collectively across national and international football authorities to deal with this serious issue, as we are also doing in this country.
"This is a matter I will take up as a priority with the new FIFA anti-racism and discrimination task force when I meet its head Jeffrey Webb in the near future."
Although progress has been made since the days when black players were regularly abused from the terraces in the 1970s and 1980s, incidences of racist chanting continue to occur.
AC Milan midfielder Kevin-Prince Boateng walked off during his team's friendly against Pro Patria after he was racially abused.
The England Under-21 side were also subjected to monkey chants during their qualifier in Serbia.
Inter's fans have been serial offenders. Inter were fined £43,000 for waving inflatable bananas and unfurling abusive banners aimed at Mario Balotelli and his team-mate Sulley Muntari in last month's Milan derby while the club also received a £12,900 fine after sections of their support sang racist chants about the former Manchester City striker during a game against Chievo.
UEFA has come in for criticism for what many deem unsatisfactory punishments concerning racism, although it did recently order Lazio to play two games behind closed doors after repeated incidents of racist chanting among their support.
Anti-racism group Kick It Out today called on European football's governing body to do more to combat the problem.
"This trend of black players representing English clubs abroad being roundly abused in this way, needs to be dealt with," Kick It Out chair Lord Herman Ouseley said in a statement released to Press Association Sport.
"Again, it appears that the match officials have failed in their duty to protect the players under their watch.
"Campaigning groups like Kick It Out, particularly in the face of domestic issues here in the UK, can only do so much.
"Unless UEFA does more on the issue, we stand still."
Tottenham fans will no doubt think that Italy in particular is struggling to deal with its battle against racism.
Spurs players were subjected to monkey-chanting against Lazio at White Hart Lane in September, and two months later during the reverse fixture fans of the Rome club sang "Juden Tottenham" - a clear anti-Semitic slur against a club with strong ties to the Jewish community in London.
Tottenham's fans have also been targeted in Rome and Lyon this season by right-wing groups.
Arsenal manager Arsene Wenger thinks incidences of racism in Italy are becoming worryingly regular.
"It looks to be a recurring problem in Italy when English teams play there and it's not acceptable," Wenger told a press conference today.
"Honestly, I don't know well enough what (UEFA) do. They came out many times to fight against it. Do we do enough? I don't know. I think it's got better. If I compare the time where I played and today, we have made big improvements but there's still some work to do and there's always the possibility of setbacks."
Wenger thinks walking off the pitch may not be the correct way to go about addressing the issue.
The Arsenal boss thinks more drastic action may need to be taken.
"I don't think that walking off the pitch is the right answer," he added.
"If you walk off, you have no game any more. When their team is losing, anybody could start racist chants and you basically get them to master the situation.
"To ban them from football or ban the clubs if you can prove they do not punish their racist fans, that is a better answer."
Tottenham goalkeeper Brad Friedel warned this morning that UEFA faces a long and hard battle in its bid to stamp out racism in football.
"(UEFA) can do what they are doing and set up campaigns and things of this nature, but until certain countries want to set up their infrastructure and school systems, and they go inside the parents' heads and the households, things will not change," the Spurs goalkeeper said.
"When they do that then things will change, not so much (through) campaigns and things of that nature.
"I think education should be more stringent."
Friedel did not hear the chanting in the San Siro last night, but he is in no doubt that racism exists throughout society.
He added: "Unfortunately I live with it every day. My wife is from Barbados so we get to see it up close and personal. People can talk about it being out of society all they want but I am afraid it's not.
"It's worldwide. I think it's disgusting. I think it's just ignorant people."