The Clarets looked to be controlling the game but deadline day signing Hudson-Odoi had other ideas, curling a stunning strike into the top corner on the hour mark.
Swiss striker Amdouni, who scored the winner here in the Carabao Cup just 20 days ago, had already given goalkeeper Matt Turner a warning sign with a stinging drive from range before netting a cool finish.
Burnley's night then ended on a sour note when top scorer Foster was sent off for violent conduct in injury time, after a needless off-the-ball elbow on Ryan Yates.
Following three straight defeats, the result allows Burnley to get off the mark this season, while Forest have made a decent start to the campaign with seven points from five games.
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