England defender Danny Rose said he was racially abused and pelted with stones in the southern city of Krusevac before, during and after Tuesday's match which England won 1-0 to qualify for next year's European Under-21 championships in Israel.
Rose has called for Serbia to be banned from internationals but the Serbian FA has denied there was racist chanting at the game and accused the player of behaving in a "vulgar manner" as a brawl broke out on the pitch after the final whistle.
A spokesman for British Prime Minister David Cameron said in a statement: "The Prime Minister is certainly aware of the events.
"He is appalled by the scenes we have seen in Serbia and we are determined to stamp out racism internationally and at home and we are giving our full backing to the FA's complaint on this issue.
"Hugh Robertson (the Minister for Sport) has written to the UEFA president urging them to investigate this issue. We would expect tough sanctions if we are going to stamp out racism from football."
Serbia have long had problems with racism and football violence. They were forced to play a Euro 2012 qualifier behind closed doors after fans rioted at an abandoned Italy game in Genoa in 2010.
"It is no good giving derisory fines as have been handed out in the past, it's not good enough that people shake hands and forget about it. We need proper sanctions," Cameron added.
England defender Rose, a Tottenham Hotspur player spending this season on loan at Sunderland, called for the strongest possible action.
"They have to be banned. I don't understand how else they can learn from it. They have to be banned," he told Sky Sports News.
England's goal, which sealed a 2-0 aggregate win, came with the last kick of the game at the end of stoppage time and fighting broke out immediately after the final whistle between players and officials as the England players were celebrating.
Rose, who was shown a red card after the final whistle for seemingly kicking the ball away in an aggressive manner, said he was racially abused throughout the evening with the insults beginning before the match even began.
"The monkey chants started straight away while we were warming up. I asked the lads if they could hear it and they said they could," he said.
"The chanting in the first half was nowhere near as bad as the second half.
"During the second half two stones hit me in the head when I went to get the ball for a throw-in. Every time I touched the ball they were doing the monkey chanting again."
The Serbian FA denied accusations of racism despite Rose's testimony and amateur video footage appearing to show hundreds of Serbian fans making monkey chants.
"The FA of Serbia absolutely refuses and denies that there were any occurrences of racism before and during the match at the stadium in Krusevac," the FSS said on its website before criticising Rose.
"Danny Rose behaved in inappropriate, unsportsmanlike and vulgar manner towards the supporters."
The English FA confirmed it had reported a number of racist incidents to the UEFA delegate with a formal protest also being lodged with European football's governing body.
"We call on UEFA to take the strongest possible action against the Serbian FA, their supporters and anyone found guilty of being involved in the numerous instances of violence and abuse," FA General Secretary Alex Horne said in a statement.
"We must question the validity of sending a team to Serbia in the future. We look forward to UEFA dealing with last night's events as a matter of urgency."
Calls for Serbia to be banned in some form also came from the likes of former England skipper Paul Ince and English Professional Footballers' Association chairman Clarke Carlisle.
The International Players' Federation FIFPro called for tougher sanctions to be imposed in cases of racism.
"FIFA, UEFA and all other confederations and federations must invoke their powers to banish these incidents," said Secretary General Theo van Seggelen in a statement.
"Racism has been a problem in football for many years. Even though FIFA, UEFA and other federations took measures to end this misbehaviour, it still occurs. Apparently, the sanctions imposed are not severe enough to eradicate this terrible conduct."
According to the Serbian B92 website, Serbia coach Aleksandar Jankovic, when asked if he felt he should apologise for the fracas, replied: "Apologise for what? It takes two for a brawl."
In June 2007, the Serbian FA was fined £16,500 because of racist chants and the behaviour of their fans at another Under-21 match against England while last year UEFA president Michel Platini said Serbia could face a ban if racism there continued.
UEFA said it was waiting for the referee's report before deciding what action would be taken.
Anti-racism campaigners have criticised UEFA in the past for what they saw as inadequate fines for monkey chants.
At Euro 2012 in June, Denmark's Nicklas Bendtner was fined 20,000 euros more for exposing an illegal advert on his underpants than Croatia were for racist chanting by their fans.