Go to the web.config system.webServer/modules section and comment/remove the elmah http module. This will prevent elmah from unhandled exception logging.
<!--<add name="ErrorLog" type="Elmah.ErrorLogModule, Elmah" preCondition="managedHandler" />-->
Log all errors into elmah manually in global.asax:
protected void Application_Error()
{
    var lastException = Server.GetLastError();
    var customException = new Exception("custom message", lastException);
    Elmah.ErrorLog.GetDefault(HttpContext.Current).Log(new Elmah.Error(customException));
}
complete web.config example:
<?xml version="1.0" encoding="utf-8"?>
<!--
  For more information on how to configure your ASP.NET application, please visit
  http://go.microsoft.com/fwlink/?LinkId=169433
  -->
<configuration>
  <configSections>
    <sectionGroup name="elmah">
      <section name="security" requirePermission="false" type="Elmah.SecuritySectionHandler, Elmah" />
      <section name="errorLog" requirePermission="false" type="Elmah.ErrorLogSectionHandler, Elmah" />
      <section name="errorMail" requirePermission="false" type="Elmah.ErrorMailSectionHandler, Elmah" />
      <section name="errorFilter" requirePermission="false" type="Elmah.ErrorFilterSectionHandler, Elmah" />
    </sectionGroup>
  </configSections>
  <appSettings>
    <add key="webpages:Version" value="3.0.0.0" />
    <add key="webpages:Enabled" value="false" />
    <add key="PreserveLoginUrl" value="true" />
    <add key="ClientValidationEnabled" value="true" />
    <add key="UnobtrusiveJavaScriptEnabled" value="true" />
  </appSettings>
  <system.web>
    <httpRuntime targetFramework="4.5" />
    <compilation debug="true" targetFramework="4.5" />
    <pages>
      <namespaces>
        <add namespace="System.Web.Helpers" />
        <add namespace="System.Web.Mvc" />
        <add namespace="System.Web.Mvc.Ajax" />
        <add namespace="System.Web.Mvc.Html" />
        <add namespace="System.Web.Routing" />
        <add namespace="System.Web.WebPages" />
      </namespaces>
    </pages>
    <httpModules>
      <!--<add name="ErrorLog" type="Elmah.ErrorLogModule, Elmah" />-->
      <add name="ErrorMail" type="Elmah.ErrorMailModule, Elmah" />
      <add name="ErrorFilter" type="Elmah.ErrorFilterModule, Elmah" />
    </httpModules>
  </system.web>
  <system.webServer>
    <validation validateIntegratedModeConfiguration="false" />
    <handlers>
      <remove name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" />
      <remove name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" />
      <remove name="ExtensionlessUrlHandler-Integrated-4.0" />
      <add name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework\v4.0.30319\aspnet_isapi.dll" preCondition="classicMode,runtimeVersionv4.0,bitness32" responseBufferLimit="0" />
      <add name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework64\v4.0.30319\aspnet_isapi.dll" preCondition="classicMode,runtimeVersionv4.0,bitness64" responseBufferLimit="0" />
      <add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" type="System.Web.Handlers.TransferRequestHandler" preCondition="integratedMode,runtimeVersionv4.0" />
    </handlers>
    <modules>
      <!--<add name="ErrorLog" type="Elmah.ErrorLogModule, Elmah" preCondition="managedHandler" />-->
      <add name="ErrorMail" type="Elmah.ErrorMailModule, Elmah" preCondition="managedHandler" />
      <add name="ErrorFilter" type="Elmah.ErrorFilterModule, Elmah" preCondition="managedHandler" />
    </modules>
  </system.webServer>
  <runtime>
    <assemblyBinding xmlns="urn:schemas-microsoft-com:asm.v1">
      <dependentAssembly>
        <assemblyIdentity name="System.Web.Helpers" publicKeyToken="31bf3856ad364e35" />
        <bindingRedirect oldVersion="1.0.0.0-3.0.0.0" newVersion="3.0.0.0" />
      </dependentAssembly>
      <dependentAssembly>
        <assemblyIdentity name="System.Web.WebPages" publicKeyToken="31bf3856ad364e35" />
        <bindingRedirect oldVersion="1.0.0.0-3.0.0.0" newVersion="3.0.0.0" />
      </dependentAssembly>
      <dependentAssembly>
        <assemblyIdentity name="System.Web.Mvc" publicKeyToken="31bf3856ad364e35" />
        <bindingRedirect oldVersion="1.0.0.0-5.2.2.0" newVersion="5.2.2.0" />
      </dependentAssembly>
    </assemblyBinding>
  </runtime>
  <elmah>
    <!--
        See http://code.google.com/p/elmah/wiki/SecuringErrorLogPages for 
        more information on remote access and securing ELMAH.
    -->
    <security allowRemoteAccess="false" />
    <errorLog type="Elmah.SqlErrorLog, Elmah" connectionStringName="elmah-sqlserver" />
  </elmah>
  <location path="elmah.axd" inheritInChildApplications="false">
    <system.web>
      <httpHandlers>
        <add verb="POST,GET,HEAD" path="elmah.axd" type="Elmah.ErrorLogPageFactory, Elmah" />
      </httpHandlers>
      <!-- 
        See http://code.google.com/p/elmah/wiki/SecuringErrorLogPages for 
        more information on using ASP.NET authorization securing ELMAH.
      <authorization>
        <allow roles="admin" />
        <deny users="*" />  
      </authorization>
      -->
    </system.web>
    <system.webServer>
      <handlers>
        <add name="ELMAH" verb="POST,GET,HEAD" path="elmah.axd" type="Elmah.ErrorLogPageFactory, Elmah" preCondition="integratedMode" />
      </handlers>
    </system.webServer>
  </location>
  <connectionStrings>
    <add name="elmah-sqlserver" connectionString="Data Source=.\SQLEXPRESS2012;Initial Catalog=Elmah;Integrated Security=True" providerName="System.Data.SqlClient" />
  </connectionStrings>
</configuration>