Index: wp-admin/includes/dashboard.php
===================================================================
--- wp-admin/includes/dashboard.php	(revision 22086)
+++ wp-admin/includes/dashboard.php	(working copy)
@@ -1237,7 +1237,7 @@
 	<h3><?php _e( 'Welcome to WordPress!' ); ?></h3>
 	<p class="about-description"><?php _e( 'We&#8217;ve assembled some links to get you started:' ); ?></p>
 	<div class="welcome-panel-column-container">
-	<div class="welcome-panel-column">
+	<div class="welcome-panel-column hide-if-no-customize">
 		<h4><?php _e( 'Get Started' ); ?></h4>
 		<p><?php _e( 'First, tweak the look of your site:' ); ?></p>
 		<a class="button-primary welcome-button" href="<?php echo add_query_arg( 'url', urlencode( admin_url( '/' ) ), wp_customize_url() ); ?>"><?php _e( 'Customize Your Site' ); ?></a>
Index: wp-admin/admin-header.php
===================================================================
--- wp-admin/admin-header.php	(revision 22086)
+++ wp-admin/admin-header.php	(working copy)
@@ -105,7 +105,8 @@
 <?php
 // If the customize-loader script is enqueued, make sure the customize
 // body classes are correct as early as possible.
-if ( wp_script_is( 'customize-loader', 'queue' ) && current_user_can( 'edit_theme_options' ) )
+if ( ( wp_script_is( 'customize-loader', 'queue' ) || 'dashboard' == $current_screen->base )
+	&& current_user_can( 'edit_theme_options' ) )
 	wp_customize_support_script();
 ?>
 
Index: wp-admin/css/wp-admin.css
===================================================================
--- wp-admin/css/wp-admin.css	(revision 22086)
+++ wp-admin/css/wp-admin.css	(working copy)
@@ -2285,6 +2285,10 @@
 		float: none;
 		width: 100%;
 	}
+	
+	.no-customize-support .welcome-panel .hide-if-no-customize {
+		display: none;
+	}
 
 	.welcome-panel .welcome-panel-column li {
 		display: inline-block;