Swift action is needed to "crush" any return of racism to football before it impacts on wider society, according to David Cameron.
The Prime Minister said he had seen for himself the influence behaviour on the pitch could exercise over young children and others off it.
He was speaking at the start of an anti-discrimination summit in Downing Street attended by leading figures in the game.
They included former England stars John Barnes and Graeme Le Saux, FA chairman David Bernstein and representatives of the Premier League and club managers.
Mr Cameron told them: "What happens on the field influences what happens off the field.
"You see children as young as six imitating the behaviour they see on the field. So this is not just important for football, it's important for the whole country."
The summit, aimed at promoting a more "inclusive" sport, comes amid fears that high-profile incidents are harming football's image.
Earlier this month Liverpool forward Luis Suarez was forced to apologise for refusing to shake hands with Manchester United's Patrice Evra before the teams' televised match at Old Trafford.
Suarez had only recently returned to action for Liverpool following a ban for racially abusing Evra during a game in October.
Just before the Suarez incident England defender John Terry was sent for trial in the summer accused of racially abusing QPR's Anton Ferdinand. Terry denies the charge.
The summit is also looking at homophobia, with Amal Fashanu also among those attending. She recently made a documentary about homosexuality in the sport, 14 years after her footballer uncle Justin hanged himself.
Mr Cameron said the UK should be proud of the "fantastic achievement" of the campaign to rid football of racism over the past two decades, noting other countries had failed to take similar action.
"But he added: "We have some problems still today (and) we need to act quickly to make sure those problems do not creep back in."
Mr Bernstein said the FA was working at all levels from the grass roots up to deal with the issue and was determined not to let complacency creep in.
The governing body had already shown its willingness to impose "very tough sanctions where necessary", he said, and would continue to do so.
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