Objective molecular dynamics and a tight-binding density functional-based model are used to investigate the mechanical stability and electronic structure of silicon nanowires with radii between 0.5 and 2 nm containing axial screw dislocations. The dislocated wires adopt twisted configurations that stabilize the dislocation at the center despite the close vicinity of surfaces, in excellent agreement with Eshelby's elasticity model of cylinders containing an axial screw dislocation. Coupled to this elasticity model, our simulations represent a new efficient method of calculating the core energy of a dislocation. We also demonstrate that the change in symmetry caused by the dislocations modulates the electronic states of the wires. The uncovered mechanical and electronic behaviors have implications for a broad class of nanomaterials grown by engaging a screw dislocation.