Material research and development is increasingly focusing on achieving specialized functionality in materials. For example, the ability to “self-heal (SH)”, or naturally repair accrued damage, is attractive because it extends the lifetime of the material by increasing resistance to damaging conditions and prolonging preservation of material properties. Additionally, shape memory (SM) materials, including SM polymers, are actively considered for their ability to change shape one or more times upon application of an external stimulus. Here, we present a polymer composite, composed of poly(vinyl acetate) (PVAc) and poly(ε-caprolactone) (PCL), exhibiting both SH and SM functionalities. In fact, the SM assists in the SH ability in a process developed by our group termed, shape memory-assisted self-healing (SMASH). The advantage of the SH composite presented here is its simple fabrication. Dual-electrospinning is used to simultaneously electrospin PVAc and PCL, achieving an interwoven polymeric composite of otherwise immiscible polymers. The dual-electrospinning method facilitates precise control of the relative weight fractions of the components, and thus allows for tuning of the material properties. Upon thermal activation, damaged PVAc–PCL composites exhibited SH under a variety of testing conditions. Furthermore, the composites exhibited impressive dual and triple SM capabilities in the dry and hydrated states, respectively. Together, the commercial availability of the components and the simplicity of preparation translate to a SMASH system that could be mass produced and used as a SH coating or alone, as a packaging material.