Using LinqPad I got this snippet working. Hopefully it solves your problem correctly.
string chart = "<div id=\"divOne\">Label.</div>;";
var regex = new System.Text.RegularExpressions.Regex(@">.*<");
var result = regex.Replace(chart, "><");
result.Dump(); // prints <div id="divOne"></div>
Essentially, it finds all characters between the opposing angle brackets, and replaces it.
The approach you take depends on how robust the replacement needs to be. If you're using this at a more general level where you want to target the specific node, you should use a MatchEvaluator. This example produces a similar result:
string pattern = @"<(?<element>\w*) (?<attrs>.*)>(?<contents>.*)</(?<elementClose>.*>)";
var x = System.Text.RegularExpressions
.Regex.Replace(chart, pattern, m => m.Value.Replace(m.Groups["contents"].Value, ""));
The pattern you use in this case is customizable, but it takes advantage of named group captures. It allows you to isolate portions of the match, and refer to them by name.