Kean, who himself left Ewood Park last September, had made the "false and defamatory" comments on a pre-season tour of the Far East in 2011. The remarks were recorded and posted on YouTube as well as being reported in newspapers.
Allardyce's lawyer Hanna Basha told the High Court on Wednesday that Kean acknowledged the allegation was completely untrue, had withdrawn it and agreed to pay a substantial and undisclosed sum in damages.
The Press Association said Kean's counsel offered his apologies for the hurt and distress caused.
Blackburn, owned by Indian poultry firm Venky's, apologised to Allardyce on their website last week.
"Whilst in Hong Kong, Mr Kean was in a bar and falsely alleged that Sam Allardyce, our previous manager, was sacked by us because he was a crook," that apology said.
"This is completely false and we apologise to Mr Allardyce for the embarrassment and distress caused to him by Mr Kean."