var container = new CookieContainer();
foreach (Cookie cookie in Driver.Manage().Cookies.AllCookies)
{
    var netCookie = new System.Net.Cookie(cookie.Name, cookie.Value, cookie.Path, cookie.Domain)
    {
        Expires = cookie.Expiry.HasValue ? cookie.Expiry.Value : new DateTime(),
        Secure = cookie.Secure
    };

    container.Add(netCookie);
}

var downloader = new CookieAwareWebClient(container);
downloader.DownloadFile(new Uri("https://dashboard-qa.sosonlinebackup.com/Accounts/ExportAccounts?_search=true&nd=1414879093707&rows=30&page=1&sidx=UserName&sord=asc&filters={\"groupOp\":\"AND\",\"rules\":[{\"field\":\"EntityFullName\",\"op\":\"eq\",\"data\":\"autest_d%20(Me:%20Distributor)\"},{\"field\":\"Mode\",\"op\":\"eq\",\"data\":\"SearchAccounts\"},{\"field\":\"EntityIdFilter\",\"op\":\"eq\",\"data\":\"3869\"},{\"field\":\"ByEntityIdSearchType\",\"op\":\"eq\",\"data\":\"CurrentEntity\"}]}"), @"d:\test.xls");

public class CookieAwareWebClient : WebClient
{
    public CookieContainer CookieContainer { get; private set; }

    public CookieAwareWebClient(CookieContainer container)
    {
        CookieContainer = container;
    }

    protected override WebRequest GetWebRequest(Uri address)
    {
        WebRequest request = base.GetWebRequest(address);
        if (request is HttpWebRequest)
            (request as HttpWebRequest).CookieContainer = CookieContainer;

        return request;
    }
}