I have a curl class, called Curl.
Let's presume i have this code:
$url = 'http://www.google.com'
$fields = array('q'=>'search term'); //maybe some other arguments. but let's keep it simple.
$curl = new Curl();
$page = $curl->post($url,$fields);
What you are describing isn't possible. For every external resource inside a web page (ie anything that's not part of the HTML content itself, such as images, scripts, stylesheets, etc), you have to make a separate request to retrieve it. This is how all browsers operate.
Many captchas work on a session basis. You initial request to the HTML page is likely creating a session cookie which would be sent back as part of the response headers. This cookie will be expected when the image is requested. If you just do a plain curl request for the image, you won't be sending that cookie, and thus you'll get a different image.
You will have to analyze the page and determine exactly what kind of session management is going on, and modify your Curl request appropriately, but as I mentioned, I suspect it'll be cookie based. You'll probably want to look at the CURLOPT_COOKIEJAR curl_setopt() parameter to get things started. You can google for pretty straightforward examples as well.