How complex 3D tissue shape emerges during animal development remains an important open question in biology and biophysics. Here, we discover a mechanism for 3D epithelial shape change based on active, in-plane cellular events that is analogous to inanimate "shape programmable" materials, which undergo blueprinted 3D shape transformations from in-plane gradients of spontaneous strains. We study eversion of the Drosophila wing disc pouch, when the epithelium transforms from a dome into a curved fold, quantifying 3D tissue shape changes and mapping spatial patterns of cellular behaviors on the evolving geometry using cellular topology. Using a physical model inspired by shape programming, we find that active cell rearrangements are the major contributor to pouch eversion and validate this conclusion using a knockdown of MyoVI, which reduces rearrangements and disrupts morphogenesis. This work shows that shape programming is a mechanism for animal tissue morphogenesis and suggests that patterns in nature could present design strategies for shape-programmable materials.