If the C++ code has to be on the client, then there is no simple way to do this for a web app. A solution may involve coding plugins for the browsers you want to support, which may then be accessed from javascript code.
If, for example, you need this for a client application, that is another case. Such a thing has been done and involves linking your application to (or running from outside) with for example chromium library, or any other javascript execution engine. That way you can create bindings to C++ classes and use such objects from javascript and vice-versa. Note that this is also not a trivial solution and may be a big effort to implement (also requires additional resources).