Gary Roberts was withdrawn after the warm-up and Rhodes, who was due to be on the bench, took advantage of his opportunity by shooting past Conrad Logan from the edge of the box after only five minutes.
Huddersfield also hit a post through a brilliant curling shot from Joey Gudjonsson in the 12th minute but were lucky to survive a grandstand finish from Rovers, who had two goals disallowed.
A long throw from James Tunnicliffe was palmed into his own net by Huddersfield goalkeeper Alex Smithies in the 83rd minute, only for the effort to be ruled out for a foul.
Then Rovers top scorer Will Hoskins shot home from close range in injury time only to be denied by a linesman's flag for offside.
Referee Brendan Malone had to be substituted with an injury after 75 minutes having earlier awarded Huddersfield a 54th-minute penalty for handball from Danny Coles. Rhodes' spot-kick was saved by the legs of Logan.