Housing Assistance for Veterans Act of 2013 or HAVEN Act - Directs the Secretary of Housing and Urban Development (HUD) to establish a pilot program to award grants to nonprofit organizations that primarily serve veterans or low-income individuals. Requires such grants to be used to rehabilitate and modify the primary residence of disabled or low-income veterans (at a specified limited or no cost to such veterans). Limits grant amounts to $1 million per organization. Requires the Secretary to direct the oversight of grant fund use. Requires a minimum of 50% matching funds by participating organizations.