Rhodes struck a 14th-minute penalty to inflict a third defeat in five matches on 10-man Charlton.
Addicks' midfielder Darel Russell conceded the spotkick and received a straight red card in the 57th minute for an apparent stamp on Antony Kay.
Huddersfield went ahead when Gary Roberts raced away down the left and cut into the box only for Russell to trip him. Rhodes fired the subsequent penalty low into the bottom corner to achieve a season's best goals tally in the post-war era.
Michael Morrison and Bradley Wright-Phillips had chances before the visitors were reduced to 10 men.
Town midfielder Kay, who escaped a first half booking, fouled Russell, who stamped back at him. Lee Novak then pushed Russell.
Referee Carl Boyeson sent Russell off and booked both Kay and Novak, while Charlton could have snatched a point in the 90th minute only for Leon Cort to head over.