Grimsby had their first shot of the second half in stoppage time, with Jason Dadi Svanthorsson denied by Glenn Morris, but the latter could not stop Rose from converting at the near post late on.
Bradford City have lost just one of their last 20 league games against Grimsby Town since 1999 (W11 D8), though it did come during the reverse fixture earlier this season in a 2-1 loss.