"I joined [the police] as a very young person in Australia, I have worked on international investigations on organised crime particularly for the last 20 years, and I think this is eminently defeatable," Eaton told Reuters.
"I believe that by good due diligence, good regulation and good oversight, we will eradicate opportunities for criminality to take over this sport," added Eaton, who worked for Interpol for more than a decade before joining FIFA.
"We can defeat it, this is not an impossible situation at all."
The sport has been hit by a recent raft of match-fixing with cases in Greece, Turkey, Italy, South Korea and Finland as well as international friendlies.
The Turkish case resulted in last season's champions Fenerbahce being withdrawn from the Champions League while Serie A side Atalanta were deducted six points in Italy.
In South Korea, the government threatened to shut down the 16-club league after 46 players were arrested in July in relation to attempts to fix 15 matches from June to October last year.
Singaporean national Wilson Raj Perumal was jailed for two years in Finland while nine players - seven Zambians and two Georgians - were given suspended sentences for trying to fix matches.
"This is an indicator of success, not an indicator of failure," said Eaton.
"The fact is, police are investigating when they weren't in the past, perhaps there's an interest in match-fixing when there wasn't before.
"This is not some sort of epidemic, this is an epidemic of enforcement, and this will help to break the back of criminal interest in football.
"It is a success story that these things are coming out, not a failure. It's dismaying for people to see but this means something is happening, it's not being ignored."
Earlier this year, FIFA set up a task force with Interpol, promising to donate 20 billion euros to the cause over the next 10 years.
UEFA President Michel Platini has described match-fixing as the biggest threat to the game.
"It's good to see the national associations such as Korea working very closely with the police and very closely with the prosecutors, and making an institutional change to the way they pay players and regulate players," Eaton said.
"We're seeing the same approach in Finland, an excellent anti-corruption approach by the Finnish federation, in fact there is a strong universal collective effort to drive criminals out of the sport."
Eaton added that the June friendly between Nigeria and Argentina, which the Africans won 4-1, was still under investigation and that he still wanted to speak to the referee Ibrahim Chaibou of Niger.
"We're trying to engage the referee, so far unsuccessfully, he retires this month but I'm endeavouring to speak to him and get his side of the story, it's not over."
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