In your form in Contact Form 7, go to “Additional settings” and paste the following:

on_sent_ok: "location = 'http://your_confirmation_page_url.com';"
on_sent_ok: "$.fancybox.close();"

This will also remove the “Message successfully sent”.

 

Sources: http://contactform7.com/redirecting-to-another-url-after-submissions/ and https://wordpress.org/support/topic/hide-success-message-after-x-seconds/