The 91-year-old said she undertook the prank as her brothers would never get their balls given back when they knocked them over the fence while playing cricket.
Dame Judi told the BBC's This Natural Life podcast: "The boys used to play cricket in the garden, and they used to be always batting the balls over into the other gardens, and there was a lady called Miss Lazenby, and she never ever would throw the ball back, or give us the balls back.
"This is a terrible story, and once the boys found a dead rat in the barn, and they parcelled it up, and they said to me, 'Judi, will you just go round and put this through Miss Lazenby's door?'.
"So I did go and squeeze it through Miss Lazenby's door, Miss Lazenby was rather red about it, saying, 'please give us our balls back', we wrote this message, it was tied on the dead rat."
She was then asked if they later got their balls back, to which Dame Judi replied: "Probably not… I don't think we were allowed to play cricket again."
Throughout her career - which has spanned seven decades - Dame Judi has won an Academy Award, a Tony Award, two Golden Globe Awards, four British Academy Television Awards, six British Academy Film Awards, and seven Olivier Awards.
She made her professional debut in 1957 with the Old Vic Company and later worked for the National Theatre Company and the Royal Shakespeare Company.
Dame Judi has also appeared as M in eight James Bond films, beginning with GoldenEye (1995) and ending with Spectre (2015), along with the series A Fine Romance and As Time Goes By, and most recently voiced The Fridge in The Magic Faraway Tree.
11°C