Trigger custom action when setting button pressed

You need a second form with admin_url('admin-post.php') as form action. Then you can hook into admin_post_custom_action to execute your action.

Sample code:

add_action( 'admin_post_wpse_79898', 'wpse_79898_test' );

function wpse_79898_test() {
    if ( isset ( $_GET['test'] ) )
        echo esc_html( $_GET['test'] );

    die( __FUNCTION__ );
}

In your settings page:

<form action="<?php echo admin_url( 'admin-post.php' ); ?>">
<input type="hidden" name="action" value="wpse_79898">
<input type="text" name="test" value="">
<?php submit_button( 'Send' ); ?>
</form>

---

Update

Here is a rather extended example. It shows:

• basic security actions (nonces, escaping),
• how to register and to use the callback,
• how to redirect back to the original page, this works even for network activated plugins,
• how to show a custom message based on a white list of allowed values.

The example I have used here – updating an option – should not be used for plugins activated for on site only. For network activated plugins though this is quite useful, because there is no options API for those.

I should add comments, but I am too lazy. 🙂 I will write a blog post about this, and update the answer later with a link.

<?php
/* Plugin Name: admin-post demo */

add_action( 'wp_loaded', array ( WPSE_79898_Admin_Post_Demo::get_instance(), 'register' ) );

class WPSE_79898_Admin_Post_Demo
{
    /**
     * Plugin instance.
     *
     * @see get_instance()
     * @type object
     */
    protected static $instance = NULL;

    protected $action     = 'wpse_79898';
    protected $option_name="wpse_79898";
    protected $page_id = NULL;

    /**
     * Access this plugin’s working instance
     *
     * @wp-hook wp_loaded
     * @return  object of this class
     */
    public static function get_instance()
    {
        NULL === self::$instance and self::$instance = new self;
        return self::$instance;
    }

    public function register()
    {
        add_action( 'admin_menu', array ( $this, 'add_menu' ) );
        add_action( "admin_post_$this->action", array ( $this, 'admin_post' ) );
    }

    public function add_menu()
    {
        $page_id = add_options_page(
            'Admin Post Demo',
            'Admin Post Demo',
            'manage_options',
            'admin-post-demo',
            array ( $this, 'render_options_page' )
        );

        add_action( "load-$page_id", array ( $this, 'parse_message' ) );
    }

    public function parse_message()
    {
        if ( ! isset ( $_GET['msg'] ) )
            return;

        $text = FALSE;

        if ( 'updated' === $_GET['msg'] )
            $this->msg_text="Updated!";

        if ( 'deleted' === $_GET['msg'] )
            $this->msg_text="Deleted!";

        if ( $this->msg_text )
            add_action( 'admin_notices', array ( $this, 'render_msg' ) );
    }

    public function render_msg()
    {
        echo '<div class="' . esc_attr( $_GET['msg'] ) . '"><p>'
            . $this->msg_text . '</p></div>';
    }

    public function render_options_page()
    {
        $option = esc_attr( stripslashes( get_option( $this->option_name ) ) );
        $redirect = urlencode( remove_query_arg( 'msg', $_SERVER['REQUEST_URI'] ) );
        $redirect = urlencode( $_SERVER['REQUEST_URI'] );

        ?><h1><?php echo $GLOBALS['title']; ?></h1>
        <form action="<?php echo admin_url( 'admin-post.php' ); ?>" method="POST">
            <input type="hidden" name="action" value="<?php echo $this->action; ?>">
            <?php wp_nonce_field( $this->action, $this->option_name . '_nonce', FALSE ); ?>
            <input type="hidden" name="_wp_http_referer" value="<?php echo $redirect; ?>">

            <label for="<?php echo $this->option_name; ?>">Enter some text:</label>
            <input type="text" name="<?php echo $this->option_name;
                ?>" id="<?php echo $this->option_name;
                ?>" value="<?php echo $option; ?>">
            <?php submit_button( 'Send' ); ?>
        </form>
        <?php
    }

    public function admin_post()
    {
        if ( ! wp_verify_nonce( $_POST[ $this->option_name . '_nonce' ], $this->action ) )
            die( 'Invalid nonce.' . var_export( $_POST, true ) );

        if ( isset ( $_POST[ $this->option_name ] ) )
        {
            update_option( $this->option_name, $_POST[ $this->option_name ] );
            $msg = 'updated';
        }
        else
        {
            delete_option( $this->option_name );
            $msg = 'deleted';
        }

        if ( ! isset ( $_POST['_wp_http_referer'] ) )
            die( 'Missing target.' );

        $url = add_query_arg( 'msg', $msg, urldecode( $_POST['_wp_http_referer'] ) );

        wp_safe_redirect( $url );
        exit;
    }
}