Such helpful. Much wow.
Thankfully I was able to solve this on my own after some intense googling since no one else seemed to have an answer. For anyone looking to accomplish the same thing, all you have to do is add the following code to the "Additional Settings" section of your CF7 form configuration:
on_sent_ok: "location = 'http://mysite.com/myform';"
Just replace the URL with the URL of your form page and presto, it'll automatically refresh back to that page when the form is submitted!