The 25-year-old was pictured undergoing a medical in Florence on Friday, and his imminent move to Serie A was acknowledged when Moyes held a press conference later in the day.
Anderson, who arrived at Old Trafford from Porto in 2007, has found first-team football hard to come by under the new United manager, starting just two league games this season.
The switch gives him a chance to impress Brazil coach Luiz Felipe Scolari ahead of the 2014 FIFA World Cup.
Moyes said: "We've not been able to get him as many games as he'd have hoped."
Anderson is set to join Fiorentina with the club fourth in Serie A and pushing for a place in the UEFA Champions League.
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