This works:
string s1 = '"five1,five2", six ,seven,"eig\\"ht1,eight2","nine",,eleven';
String lst = s1.replaceAll('\\\\"','\\$');
System.debug(lst);
See e.g. Can't escape the backslash with regex? for some detail on needing multiple levels of escaping.
But the quick explanation is that for a backslash to be treated literally by the Apex compiler it needs to be escaped itself by a backslash, and then another layer of escaping is needed for regex engine too. Hence 2 * 2 = 4 backslashes needed.
And $
has special meaning in regex too, so that character needs escaping.