Fulham had the better chances in the first-half but were unable to get past Illan Meslier.
The second-half saw little chances from both sides and the game eventually went to the penalties.
Meslier was once again the hero as he saved from Rodrigo Muniz to win it for the Premier League side in sudden death.
Marcelo Bielsa will be hoping Leeds can carry some momentum into Saturday's clash with West Ham.
The Yorkshire side have drawn three and lost two from their opening five Premier League games.
Bielsa said after the game: "It was difficult for us. We conceded too many chances to the opponent. It was difficult for all our players.
"The young players showed personality and character. Joe Gelhardt had missed his last two penalties and took on board the responsibility and managed to score and Stuart McKinstry also had to take one in the most decisive moments and they showed character."
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