Mechanical performance of biological structures, such as tendons, byssal threads, muscles, and spider webs, is determined by a complex interplay between material quality (intrinsic material properties, larger scale morphology) and proximate behaviour. Spider orb webs are a system in which fibrous biomaterials--silks--are arranged in a complex design resulting from stereotypical behavioural patterns, to produce effective energy absorbing traps for flying prey. Orb webs show an impressive range of designs, some effective at capturing tiny insects such as midges, others that can occasionally stop even small birds. Here, we test whether material quality and behaviour (web design) co-evolve to fine-tune web function. We quantify the intrinsic material properties of the sticky capture silk and radial support threads, as well as their architectural arrangement in webs, across diverse species of orb-weaving spiders to estimate the maximum potential performance of orb webs as energy absorbing traps. We find a dominant pattern of material and behavioural coevolution where evolutionary shifts to larger body sizes, a common result of fecundity selection in spiders, is repeatedly accompanied by improved web performance because of changes in both silk material and web spinning behaviours. Large spiders produce silk with improved material properties, and also use more silk, to make webs with superior stopping potential. After controlling for spider size, spiders spinning higher quality silk used it more sparsely in webs. This implies that improvements in silk quality enable 'sparser' architectural designs, or alternatively that spiders spinning lower quality silk compensate architecturally for the inferior material quality of their silk. In summary, spider silk material properties are fine-tuned to the architectures of webs across millions of years of diversification, a coevolutionary pattern not yet clearly demonstrated for other important biomaterials such as tendon, mollusc byssal threads, and keratin.