We run two parallel stacks, a staging stack and a production stack. New versions are automatically deployed to Staging on every commit as part of CI.
Note that you don't need to run database migrations manually for the Elastic Beanstalk environments. They happen automatically. See this ebextension for details.
Visit the list of application versions. Under the "Deployed to" column, you should see both prod and stage next to some version. Hopefully stage is at or ahead of prod! If staging looks good (TODO: Help define what this means), go ahead and hit the checkbox next to the staged version, open the "Actions" menu, select "Deploy" and deploy to prod. That will kick off the migration, and if successful, the prod indicator should move to align with stage.