The match ended in chaotic scenes as Connor Wickham's late winner sparked a fracas that resulted in Rose being sent off, with players and staff from both sides clashing on the pitch.
Mr Cameron called for tough sanctions to be imposed upon the Serbian FA, with the Football Association making an official complaint to Uefa, urging the European governing body to stamp out racism.
A spokesan for the PM said: "[Mr Cameron] was appalled by the scenes that we have seen in Serbia.
"Clearly it is for Uefa to investigate this issue but we would expect tough sanctions. If we are going to stamp out racism from football, then it is no good giving derisory fines, as have been handed out in the past.
"It is not good enough to say that people should shake hands and forget about it.
The statement confirmed the government would support the FA's official complaint, saying: "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."
Sports minister Hugh Robertson has also written a letter to Uefa president Michel Platini, insiting racism has no place in society or sport.
"The British government stands absolutely side by side with the FA in finding what happened last night absolutely appalling," Robertson told Sky Sports News.
"Our young players were subject to extraordinary levels of racism and provocation that have no place in society as a whole, let alone in football. We absolutely want to make sure that Uefa realise this is serious and take the necessary action."
Referring to Serbia's £16,500 fine for racist chanting from their fans during the 2007 clash between the two sides, Robertson was unequivocal in his demand for a tougher punishment.
He said: "I don't want to see some derisory fine this time around. It is no good imposing some small fine, it doesn't work."