Can you use curl to send a login post request to wordpress?

Solution: 

I think WordPress don’t let you to do this because of missing security nonce or referrer page.

My suggestion is to install/develop a plugin that supports remote login and then send the login request to your website based on the plugin structure.

You can use following function for auto login

function auto_login( $username ) {
    // log in automatically
    if ( !is_user_logged_in() ) {
        $user = get_userdatabylogin( $username );
        $user_id = $user->ID;
        wp_set_current_user( $user_id, $user_login );
        wp_set_auth_cookie( $user_id );
        do_action( 'wp_login', $user_login );
    }     
}

Then what you need is to just call it as following:

auto_login('admin');

Hope it helps.