How do I manually upgrade my WordPress theme?

This question has been frustrating me for the past few days. In most cases, it's a very simple process that's managed through the WordPress theme directory. After you find and activate the theme you like, WordPress simply notifies you when there is an update. All you have to do then is tell it to install the update and the rest is taken care of.

However, that doesn't work if your theme is not in the WordPress directory. I recently updated from a free theme that's in the directory to a premium version of the same theme, which the creators sell on their website. The link to download it was sent in an email, and it came with instructions on how to upload it through the WordPress dashboard. This is how that part works:
Navigate to "Appearance>Themes" from the side menu of your WordPress dashboard
At the top, left corner of the page, click the "Add New" link
At the top, left corner of the next page, click the "Upload Theme" link
On the next page, there is a place to select a .zip file of the theme you've acquired. Choose that file from the location where it's stored on your computer and click the button to upload/instal it.
After that, make sure the theme is activated and then do any required setting changes.
That should all work fine, at least it did for me when I first got the premium theme. Now, here comes the frustrating part. If you get an updated version of your theme, like I did (They notified me by email that there was an update with new features), the previous upload instructions do not work. When WordPress recognizes that the theme directory is already installed, it causes the update to stop.
After asking about this and doing some research, I found a simple solution for this problem that doesn't require deleting files through FTP and works for self installed plugins as well. It's a plugin called "Easy Theme and Plugin Upgrades". It allows you to upload your theme or plugin upgrade with the same instructions I explained above and will do all the work for you.
I know most people won't run into this problem but, since I did, I wanted to share what I found. Hopefully it will help the small few who will inevitably be searching for a solution like I was.