You should edit the following files:
design/themes/YOUR_THEME/mail/templates/orders/order_notification.tpl
design/backendmail/templates/orders/order_notification.tpl
Replace:
<b>{$order_header}:</b><br /> {include file="orders/invoice.tpl"}
with something similar to:
{if $order_info.status == 'P' || $order_info.status == 'C'} <b>{$order_header}:</b><br /> {include file="orders/invoice.tpl"} {/if}
Do not forget to clear cache