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Charlton shaded possession in the early stages but it was Blackburn who had the first strike on goal when Simon Vukcevic fired wide from range.
Danny Murphy also failed to hit the target with a free kick as Charlton restricted the home side to efforts from distance.
Charlton burst forward on the counter and took the lead midway through the first half when Dale Stephens slammed into the top corner from just outside the area.
Blackburn went close soon after when Colin Kazim-Richards had a strong header well saved before Charlton scrambled clear a dangerous corner.
The home side equalised quickly after the interval when Rhodes reacted quickest to a Kazim-Richards cross to finish from close range.
Appleton sent Nuno Gomes on in the closing stages but he could not find an equaliser as Charlton held on to keep their 100% league record in 2013 alive.