Midfielder Cavanagh was shown a straight red card by referee Seb Stockbridge in the seventh minute following a high challenge on midfielder Robbie Weir.
Dale manager Keith Hill felt Cavanagh should have been shown a yellow card for an earlier challenge on Weir, which resulted in treatment for an eye injury, but that the second challenge was accidental.
Hill said after his side's 1-0 defeat: "I'm relatively pleased by what the players are doing, what I'm trying to achieve and what the club is trying to achieve, but I can't legislate for match-changing decisions that the referee makes so early in the game.
"It has been a decision that has somewhat dictated the remainder of the game and that is unfortunate."
Cavanagh will sit out an automatic three-game ban if Dale's appeal is unsuccessful and miss Saturday's home game against Chesterfield and successive away fixtures at Plymouth and Oxford.