The SMTP module alone will not route system generated emails from Ubercart through the SMTP module. In order to accomplish this you also need the MailSystem module.
Then, you need to configure the mail system module with a new class that users Ubercart to format emails and SMTP to send the emails.
Lastly, change all of the Ubercart settings to use the newly generated class:
Thanks to this post for the original fix:
https://www.drupal.org/node/2298963