Certain aspects and embodiments of the present invention are directed to a subsurface safety valve that can be disposed in a wellbore that is through a fluid-producing formation. The subsurface safety valve can include a closure mechanism, a sleeve, and a control line. The closure mechanism can be positioned in a passageway defined by a tubing string. The closure mechanism can be configured to prevent a flow of fluid to a portion of the passageway that is closer to a surface of the wellbore than the closure mechanism. The sleeve can be positioned in the passageway adjacent to the closure mechanism. The control line can communicate pressure to a piston from a pressure source within an inner diameter of the tubing string, causing the piston to apply a force to the sleeve. The sleeve can open the closure mechanism in response to the force being applied to the sleeve.