Collections of DNA sequences can be rationally designed to self-assemble into predictable three-dimensional structures. The geometric and functional diversity of DNA nanostructures created to date has been enhanced by improvements in DNA synthesis and computational design. However, existing methods for structure characterization typically image the final product or laboriously determine the presence of individual, labelled strands using gel electrophoresis. Here we introduce a new method of structure characterization that uses barcode extension and next-generation DNA sequencing to quantitatively measure the incorporation of every strand into a DNA nanostructure. By quantifying the relative abundances of distinct DNA species in product and monomer bands, we can study the influence of geometry and sequence on assembly. We have tested our method using 2D and 3D DNA brick and DNA origami structures. Our method is general and should be extensible to a wide variety of DNA nanostructures.