To add a custom router class to the list of routers in Magento, the routes.xml file is used. This file should be located in the etc directory of the module, under the appropriate area (either frontend or adminhtml). Within the routes.xml file, you define a router with an ID, a route with an ID and frontName, and specify the module that the route corresponds to. This setup allows Magento to recognize and utilize the custom router when processing URLs, directing requests to the appropriate controllers based on the custom routing logic defined.