The Tigers led 2-0 inside the opening eight minutes at Wembley Stadium before the Gunners fought back to win 3-2 after extra time.
While he hopes that Hull's fans are proud of the players' efforts, Rosenior insists that the result will be playing on his mind for some time yet.
"I'm proud to play for this football club because it's a special one and the fans are fantastic," the 29-year-old is quoted as saying by Sky Sports News.
"Hopefully we've done them proud. When the time comes I'm sure I'll enjoy this too but at the moment it still hurts."
Hull finished the Premier League season in 16th place in the table.