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.
‘The pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
