The Argentine also won the best goal award for his spectacular backheel during a four-goal haul against Sampdoria last March.
Icardi disagreed with the choice, insisting his "one (goal) at AC Milan was more beautiful".
"It's a group of great men, with a great club behind us," said Allegri, who has won the award four times.
"I'm fortunate to have worked with so many marvellous players and tonight I see the maestro Andrea Pirlo too." Allegri's side are on track for an eighth straight Serie A title, with Allegri hailing the impact of five-time Ballon d'Or winner Cristiano Ronaldo who arrived last summer.
"Ronaldo has brought professionalism, competitiveness and is a humble guy," said Allegri.
Meanwhile, Gianluca Rocchi won the award for the best referee 24 hours after being whistled off the pitch at the Stadio Olimpico over his decision not to award Roma a penalty in a 2-2 draw against Inter Milan.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)