Brown and Kinnear squared up to each other and were subsequently sent to the stands by referee Phil Dowd.
Hull won the match 1-0 courtesy of a Cousin goal eight minutes from time.
A statement from the FA read: "At a regulatory commission hearing today, Hull City manager Phil Brown was warned as to future conduct and fined £2,500.
"Brown admitted a charge of improper conduct relating to a touchline confrontation involving himself and Joe Kinnear during an FA Cup match at St James' Park on 14 January."
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