The Europa Conference League final was goalless when Biraghi went to take a corner at the end of the Eden Arena in Prague where West Ham fans were sitting.
He required a head strapping in order to continue, after referee Carlos Del Cerro Grande had briefly called a halt to the game and a stadium announcement had issued a warning for fans to stop throwing things and respect the players and officials before the match could continue.
West Ham issued a statement promising to take action against anyone found to have committed an offence and ban them from attending matches at the London Stadium.
A statement from the Premier League club read: "West Ham United unreservedly condemn the behaviour of a small number of individuals who threw objects onto the pitch during tonight's UEFA Europa Conference League fina.
"These actions have no place in football, and do not in any way represent the values of our football club and the overwhelming majority of our supporters, who have behaved impeccably in Prague this week and throughout our last two seasons in European competition.
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"We will work with the police and other stakeholders to review the incidents and act against anyone found guilty of an offence.
"In line with our zero-tolerance approach, anyone identified will have their details passed to the police and will be given an indefinite ban and therefore be unable to enter London Stadium and travel with the club. There is no place for this kind of behaviour at West Ham United."
Speaking on talkSPORT, Sam Matterface condemned the action, saying live on air: "This is absolutely mindless. Paqueta, Coufal and Aguerd have gone over to the West Ham fans and said, 'calm down'.
"Michail Antonio is going over there and going, 'do not do it'. It's causing serious issue, this is now going to damage the reputation of West Ham in Europe.
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"They've been brilliant over the course of the week, the atmosphere has been jubilant - party-like - rather than aggressive.
"And then tonight, we've had a series of incidents where they have been throwing plastic cups onto the pitch and one of them now has been thrown with such velocity that it's cut the head of the Fiorentina captain."
Alongside him was ex-Hammers star Dean Ashton, who said: "It's just disgusting to see. I don't see why they do it.
"It's been a great week, so far it's been an excellent game, really contested by the two sides.
"Declan Rice has gone over to try and console Biraghi and show that he's supporting him and he doesn't condone those actions."
It was not the first time that the Italian had objects from the crowd thrown at him.
This was the third corner that Fiorentina had from the right-hand corner flag.
And every time, fans were throwing cups at the 30-year-old, though this was the first time that one connected.
After he was struck, Biraghi sarcastically applauded the fans behind the corner before he was given medical treatment.
Ashton added: "There's the odd idiot who thinks this is the right thing to do. It's horrible as a player fearing for your safety. It's just mindless. We hate to see it."
It has been a feisty atmosphere inside the Fortuna Arena in Prague, where both sets of fans were only given around 5,000 tickets in the 20,000-capacity ground.
There had been trouble in the city before a ball had even been kicked with chairs being thrown between fans.
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West Ham went on to win the Conference League when Jarrod Bowen scored a 90th-minute winner in dramatic circumstances.
Said Benrahma had scored a penalty before Giacomo Bonaventura netted a beauty but it was not enough for Fiorentina as Bowen sent the Hammers wild.