The Terriers’ frontman had already been denied twice by the Brunton Park woodwork on Tuesday night, but got his reward when he was left with a simple finish from substitute Lee Novak’s cross after a slip by Carlisle defender James Chester.
Carlisle came closest to opening up a solid Huddersfield defence in the 36th minute when Ian Harte’s free-kick prompted a scramble during which James Berrett and Gary Madine had close-range blasts blocked.
Rhodes was the man to watch in the visitors’ attack, but must have thought his luck was out when he hit the crossbar twice in a three-minute spell in the second half.
The first effort was a header as keeper Adam Collin struggled to get back after a slip, and then Rhodes lashed a thumping 25-yarder which cannoned back off the woodwork.
However, Rhodes had the final say as extra time loomed, converting the easiest of opportunities from eight yards with the home defence split wide apart.