include_once( ABSPATH . 'wp-admin/includes/plugin-install.php' ); //for plugins_api..

$plugin = 'plugin-name';

$api = plugins_api( 'plugin_information', array(
    'slug' => $plugin,
    'fields' => array(
        'short_description' => false,
        'sections' => false,
        'requires' => false,
        'rating' => false,
        'ratings' => false,
        'downloaded' => false,
        'last_updated' => false,
        'added' => false,
        'tags' => false,
        'compatibility' => false,
        'homepage' => false,
        'donate_link' => false,
    ),
));

//includes necessary for Plugin_Upgrader and Plugin_Installer_Skin
include_once( ABSPATH . 'wp-admin/includes/file.php' );
include_once( ABSPATH . 'wp-admin/includes/misc.php' );
include_once( ABSPATH . 'wp-admin/includes/class-wp-upgrader.php' );

$upgrader = new Plugin_Upgrader( new Plugin_Installer_Skin( compact('title', 'url', 'nonce', 'plugin', 'api') ) );
$upgrader->install($api->download_link);