The Addicks were comfortable winners in Saturday's League One clash at Boundary Park but results elsewhere stopped them leaping from fifth place to second as Leeds secured the second promotion place.
Phil Parkinson's side will now meet Swindon for a place at Wembley to face the winner of Millwall's semi-final against Huddersfield.
After a slow start Charlton went in front midway through the first half when Lloyd Sam's deflected cross was tapped in by Nicky Bailey.
Miguel Llera hit a tame fee-kick from outside the box yet it somehow squirmed through Darryl Flahavan's grasp to leave the Oldham keeper red-faced for Charlton's second goal.
The visitors cruised through to the final whistle, securing their ninth away win of the season as they attempt to bounce straight back up to the Championship at the first attempt, while Oldham, who sacked manager Dave Penney in midweek, posed little threat apart from a Dale Stephens shot which was well saved by Darren Randolph.