Emmanuel Adebayor was the main target of abuse from Inter fans in the San Siro, with the Togo striker scoring the extra-time goal that saw Spurs advance on the away goals rule after a 4-1 defeat on the night left the tie level at 4-4.
Despite ongoing attempts by UEFA and FIFA to eradicate racism in football, the problem persists and Bernstein admits the regularity of such abuse is a concern.
"I condemn the racist abuse that Tottenham Hotspur players suffered at last 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."
Inter Milan have been in trouble in the past for the actions of some sections of their support.
They were fined £43,000 for waving inflatable bananas and unfurling abusive banners aimed at Mario Balotelli and team-mate Sulley Muntari in last month's Milan derby.
The club was also handed a £12,900 fine after sections of their support directed racist chants towards Balotelli during a game against Chievo.
UEFA has been criticised in some quarters for what many regard as unsatisfactory punishments concerning racism, although it did recently order Lazio to play two games behind closed doors in the wake of repeated incidents of racist chanting among their support.
But Anti-racism group Kick It Out has called on European football's governing body to step up their efforts to combat the problem.
"This trend of black players representing English clubs abroad being roundly abused in this way, needs to be dealt with," said Kick It Out chair Lord Herman Ouseley.
"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."