The Uruguayan was charged with violent conduct after the FA reviewed video footage of the incident during PNE’s 1-0 defeat at the Riverside Stadium on Saturday.
Footgae showed Stuani swing an elbow into Cunningham’s head as they challenged to meet a corner.
Referee Oliver Langford and his assistants missed it at the time but it was picked up on camera.
Middlesbrough contested the charge but a disciplinary panel found the case proven and the three-game ban was imposed on Stuani, the first of which he served on Tuesday night when Boro beat Reading 2-1 to go top of the Championship.
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The disciplinary process saw three former referees watch footage of the incident before the FA chose to charge Stuani.
A statement from the FA said: “A regulatory commission found the case proven and Stuani will serve an immediate three-match suspension.”
Giving his reaction to the ruling, Middlesbrough head coach Aitor Karanka said: “I don’t want to say a lot of things but the only thing I can say is that my feeling is I don’t think it’s fair because I don’t think all the teams are playing in the same conditions.”