Customizing comment form in WordPress theme

I want to customize the HTML for the comments form in my WordPress theme. After reading the documentation (http://ift.tt/Lj5rWD) it seems that you can customize some of the fields but not the form as a whole. For example, I want to add to the class attribute of the comments form itself, and it seems there is no way to do this.

For the search form, you can simple create a template searchform.php and put in the custom HTML in this form. Is there any similar approach that can be used for the comments form?