I had updated the customErrors node in web.config as below :
<customErrors defaultRedirect="~/ErrorPage.aspx" mode="RemoteOnly" />
but the custom error page doesn't seem to show up on every error. Instead, the ASP.NET error pages bubble up. How to get all errors to redirect to my custom error page?

Sorry, I changed it to :

    <customErrors defaultRedirect="~/ErrorPage.aspx" mode="On">
                <error statusCode="403" redirect="~/ErrorPage.aspx" />
                <error statusCode="404" redirect="~/ErrorPage.aspx" />
    </customErrors>

But it still didn't work. Even the 404 directive doesn't seem to work. Is it an issue of IIS not handing over the processing to ASP.NET? Or did I miss some other DNN/web.config setting?

No, no this has nothing to do with ASP, we're talking DNN on .NET aren't we? I didn't do any IIS mappings at all. And even if I wanted to, what if I don't have access to IIS?

From what you're saying, it looks like all types of exceptions cannot be caught, after all, but I'm not sure whether its because of DNN or .NET.