As of WordPress 4.6.0 there is a resource hints API that automatically adds all unique enqueued domains, which you can override with wp_resource_hints - you should only use the following answer if you're stuck with < 4.6.0
All you can do is bump the priority of your hook:
add_action( 'wp_head', 'dns_prefetch', 0 /* Highest priority */ );
but does this put my code just after the start of tag?
No, but it will output before any other function attached to wp_head i.e. other <link />'s, most stylesheets, plugin scripts etc.
To put it right after <head> you'll most likely need to edit your theme's header.php directly.
is there really no other way around?
You can hack it with a bit of output buffering:
function wpse_177395_start_buffering( $template ) {
ob_start();
return $template;
}
add_filter( 'template_include', 'wpse_177395_start_buffering' );
function wpse_177395_flush() {
$content = ob_get_clean();
$content = preg_replace( '/<head[^>]*>/', '$0
<link rel="dns-prefetch" href="//cdn.mysite.com" />
<link rel="dns-prefetch" href="//fonts.googleapis.com" />',
$content
);
echo $content;
}
add_action( 'wp_head', 'wpse_177395_flush', 0 );