The 18-year-old, on as a half-time substitute, converted from close range after an Alan Quinn shot had been charged down in the 73rd-minute.
The Bluebirds appealed in vain for offside as Rhodes celebrated his first League goal which equalised Gavin Rae’s opener.
But the home side were furious not to get all three points after a 90th-minute Danny Haynes effort was ruled out.
Haynes drilled the ball into the roof of the net from 12 yards but referee Russell Booth decided there had been a foul as the ball was crossed.
Cardiff took the lead in the 37th minute through Rae’s fourth goal of the season.
Teenage winger Aaron Ramsey tricked his way past Alex Bruce and crossed for Rae to convert from two yards.